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Granular Instrumental Variables: Estimation and Inference
\title{Granular Instrumental Variables: Estimation and Inference}
\author{Jinyong Hahn\thanks{
Department of Economics, UCLA, Los Angeles, CA 90095-1477 USA. Email:\
[email removed]} \\
UCLA \and Niu He\thanks{
Department of Economics, UCLA, Los Angeles, CA 90095-1477 USA. Email:\
[email removed]} \\
UCLA \and Zhipeng Liao\thanks{
Department of Economics, UCLA, Los Angeles, CA 90095-1477 USA. Email:\
[email removed]} \\
UCLA \and Wenyu Zhou\thanks{
International Business School, Zhejiang University, Haining, Zhejiang
314400, China. Email: [email removed].} \\
Zhejiang University}
\date{\today }
\maketitle
\begin{abstract}
We develop an estimation and inference framework for granular instrumental variables (GIVs) in models with latent aggregate shocks. Our key insight is that valid GIVs are characterized by the orthogonal complement of the factor-loading space. This characterization yields a feasible procedure for constructing GIVs when factor loadings are unknown and does not require a large cross-sectional dimension. We provide practical procedures for inference and specification testing, and apply the framework to estimate the aggregate equity market multiplier. Our empirical results reveal substantial heterogeneity in equity demand elasticities across investor sectors and may provide nuanced support for the inelastic-markets hypothesis.
\bigskip
\noindent JEL Classification: C13, C26, C51, G12\bigskip
\noindent \textit{Keywords:} Granular Instrumental Variables; Latent
Aggregate Shocks; Identification and Inference; Asset Demand; Market
Elasticity.
\end{abstract}
\section{Introduction\label{sec:intro}}
Understanding how aggregate outcomes respond to shocks is a central
objective in economics and finance. In many applications, researchers
observe a large cross section of entities that are simultaneously exposed to
a small number of common shocks. Examples include firms responding to
aggregate demand conditions, financial institutions adjusting portfolios in
response to market forces, and countries reacting to global macroeconomic
shocks. A common feature of these environments is that the variables of
interest are jointly determined in equilibrium, creating endogeneity
problems that complicate identification and estimation of structural
parameters.
Recently, \cite{gabaix2024granular} proposed a novel identification strategy
based on granular instrumental variables (GIVs). The approach builds on the
insight from the granularity literature that when a small number of firms,
industries, countries, investors, or borrowers account for a non-negligible
share of aggregate activity, idiosyncratic shocks to these units may survive
aggregation and influence aggregate outcomes.\footnote{
See, among others, \cite{gabaix2011granular,
acemoglu2012network,di2014firms, baqaee2019macroeconomic,gaubert2021granular}
.} Exploiting this feature, GIV extracts the idiosyncratic component of
observables after controlling for common factors and aggregates these
components using size weights to construct instruments for causal parameters
such as elasticities and multipliers. Under suitable restrictions on the
covariance structure of the idiosyncratic shocks, the resulting instruments
are orthogonal to equilibrium disturbances and can therefore identify
structural parameters. Unlike earlier approaches that use idiosyncratic
shocks to variables excluded from the estimating equation,\footnote{
See, for instance, \cite{leary2014peer,amiti2018much,amiti2019international}.
} GIV constructs instruments from the idiosyncratic component of the
variables entering the estimating equation itself. As a result, the
methodology does not rely on traditional excluded instruments, which are
often difficult to justify or unavailable in practice.
Several recent papers extend the baseline GIV framework.
\citet{banafti2022inferential} study inference in large-$n$, large-$T$
settings with unknown factors and loadings. \citet{baumeister2023uncovering}
develop a likelihood-based approach, while \citet{qian2023heterogeneity}
allows for heterogeneous spillovers. GIV has also become an important
identification device in empirical macroeconomics and finance, including
applications to stock-market demand, exchange rates, bank lending, and asset
pricing.\footnote{
Recent applications include \citet{galaasen2020granular,
camanho2022global,ma2022mutual,dong2025fast}; many others are discussed in
\citet{gabaix2024granular}.}
Despite its growing importance, several econometric questions remain
unresolved. The central challenge is that valid GIV construction requires
knowledge of the factor-loading space associated with latent aggregate
shocks, which is rarely observed in practice. Existing implementations
therefore estimate latent factors and construct GIVs in a second step, a
strategy that typically relies on a large cross-sectional dimension and
strong normalization assumptions. Moreover, little is known about the
consequences of estimating the factor-loading space for identification,
estimation, and inference.
This paper develops an estimation and inference framework for structural
models identified by GIVs when the factor-loading space is unknown. Rather
than estimating latent factors and constructing instruments in a second
step, we show that the relevant GIV space can be recovered directly from the
covariance structure of the observables. Specifically, the admissible GIVs
are generated by the orthogonal complement of the factor-loading space,
which can be identified from the eigenspace associated with the smallest
eigenvalues of the covariance matrix. This insight transforms the
construction of GIVs into a covariance-based problem and yields a feasible
procedure for constructing instruments directly from the data. The resulting
estimator remains valid even when the number of entities is fixed and
therefore does not require the cross-sectional dimension to diverge with the
sample size.
The characterization also provides a transparent identification strategy. We
show that all admissible GIVs are generated by the orthogonal complement of
the column space spanned by the factor loadings and the vector of ones. When
the factor-loading space is unknown, the relevant orthogonal complement can
be consistently recovered from the eigenspace associated with the smallest
eigenvalues of the covariance matrix of the observables. This result yields
a feasible GIV estimator and forms the basis for inference with estimated
GIVs.
Building on this characterization, we establish consistency and asymptotic
normality of the feasible GIV estimator and develop practical procedures for
inference and specification testing. In particular, we derive feasible
standard errors, establish the asymptotic validity of an over-identification $
J$-test when the GIVs are estimated, and propose a BIC-type criterion for
determining the dimension of the factor-loading space. Monte Carlo
simulations show that the feasible estimator performs similarly to an oracle
estimator that knows the true factor-loading space and that the proposed
inference procedures perform well in finite samples.
Beyond estimation and inference, the paper clarifies several identification
issues that arise when factor loadings are unknown. We show that certain
restrictions commonly interpreted as normalizations on the factor loadings
instead impose substantive restrictions on the latent factors. We also
examine the identification strategy in \citet{gabaix2024granular} and show
that the moment conditions in their Proposition~7 may fail to identify
the structural parameters when factor loadings are unknown. These findings
highlight the challenges of identification in the presence of latent
aggregate shocks and motivate the alternative characterization developed in
this paper. The proposed framework avoids these restrictions and extends
naturally to settings with additional exogenous regressors, unbalanced
panels, and heterogeneous demand elasticities.
Finally, we return to the estimation of the aggregate equity market multiplier in demand-based asset pricing \citep{gabaix2021search}. This application is particularly relevant because the factor-loading space is unknown and only twelve investor sectors are available in the data. As a result, the large-cross-section justification underlying existing GIV procedures is difficult to invoke directly, making the setting a natural environment in which to assess the practical importance of estimating GIVs when factor loadings are unobserved. Applying our framework, we obtain estimates and conduct inference for the aggregate multiplier together with a formal specification test of the underlying GIV moment conditions. The empirical results provide evidence consistent with highly inelastic aggregate equity demand.
The remainder of the paper is organized as follows. Section~\ref{sec:
S_model} introduces the main ideas in a simplified framework. Section~\ref
{sec: G_model} develops the general model, establishes identification, and
presents estimation and inference procedures based on estimated GIVs.
Section~\ref{sec: Extension} studies identification when factor loadings are
unknown, demonstrates a failure of identification in the moment conditions
proposed by \citet{gabaix2024granular}, and develops extensions to models
with exogenous regressors, unbalanced panels, and heterogeneous demand
elasticities. Section~\ref{sec: MC} reports Monte Carlo evidence, Section~
\ref{sec:emp} presents an empirical application to the aggregate equity
market multiplier, and Section~\ref{sec:conclusion} concludes. Proofs and
additional technical and empirical results are collected in the Online
Appendix.
\textit{Notation.} We use $K$ to denote a generic strictly positive constant that may vary from place to place but does not depend on the sample size $T$. We write $a\equiv b$ to indicate that $a$ is defined as $b$. For any positive integer $k$, let $\mathbf{I}_k$, $\mathbf{1}_k$, and $\mathbf{0}_k$ denote the $k\times k$ identity matrix, the $k\times1$ vector of ones, and the $k\times1$ vector of zeros, respectively. For any vector $x_t\in\mathbb{R}^n$ (possibly indexed by $t$) and any weight vector $\Greekmath 0121 \in\mathbb{R}^n$ satisfying $\Greekmath 0121 ^\top\mathbf{1}_n=1$, define the weighted average $x_{\Greekmath 0121 ,t}\equiv\Greekmath 0121 ^\top x_t$. For any matrix $A$, let $\func{col}(A)$ and $\func{rank}(A)$ denote its column space and rank, respectively. We use $\Vert A\Vert$ and $\Vert A\Vert_{\mathrm{o}}$ to denote the Frobenius norm and operator norm of $A$, respectively, and $M_A$ to denote the orthogonal projection matrix onto the orthogonal complement of $\func{col}(A)$. For any square matrix $A$, let $\Greekmath 011A _{\min}(A)$ and $\Greekmath 011A _{\max}(A)$ denote its smallest and largest eigenvalues, respectively. For any two matrices $A$ and $B$, let $\mathrm{diag}(A,B)$ denote the block-diagonal matrix with $A$ and $B$ on its main diagonal, and let $A\otimes B$ denote their Kronecker product. Finally, for any square matrix $A$, let $\func{vech}(A)$ denote the half-vectorization of $A$, obtained by stacking the elements of its lower triangular part (including the diagonal) column by column.
\section{A Simplified Framework \label{sec: S_model}}
We first illustrate the intuition of GIV using the following simplified
model:
\begin{align}
y_{i,t} & =\Greekmath 011E p_{t}+\Greekmath 0111 _{t}+u_{i,t}, \label{S_demand} \\
p_{t} & =\Greekmath 0120 y_{S,t}+\Greekmath 0122 _{t}, \label{S_supply}
\end{align}
where $y_{i,t}$ denotes the log demand of entity $i\in \{1,\ldots,n\}$ at
time $t$, and $p_{t}$ is the log price common to all entities. The latent
variables $\Greekmath 0111 _{t}$ and $u_{i,t}$ with $\mathbb{E}[u_{i,t}]=0$ represent
aggregate and idiosyncratic demand shocks, respectively. The parameter $\Greekmath 011E $
measures the demand elasticity. Equation~(\ref{S_supply}) describes the
supply side, where $y_{S,t}\equiv S^{\top}y_{t}$ denotes aggregate demand, $
S\equiv(s_{i})_{i\leq n}$, and $y_{t}\equiv(y_{i,t})_{i\leq n}$, with $s_{i}$
denoting the market share of entity $i$. The term $\Greekmath 0122 _{t}$ is the
supply shock, and $\Greekmath 0120 $ denotes the supply elasticity. Although stylized,
this model is widely used in the macroeconomics and finance literature
\citep{gabaix2021search,camanho2022global}.
Let $u_{t}\equiv(u_{i,t})_{i\leq n}$. The following assumptions are
maintained throughout this section:
\begin{equation}
\mathrm{Cov}(\Greekmath 0122 _{t},u_{t})=\Greekmath 011B _{\Greekmath 0122 u}\mathbf{1}^{\top}
_{n},\qquad \mathrm{Cov}(\Greekmath 0111 _{t},u_{t})=\Greekmath 011B _{\Greekmath 0111 u}\mathbf{1}^{\top}
_{n},\qquad \mathrm{Var}(u_{t})=\Greekmath 011B _{u}^{2}\mathbf{I}_{n},
\label{GIV_Validity}
\end{equation}
where $\Greekmath 011B _{\Greekmath 0122 u}$ and $\Greekmath 011B _{\Greekmath 0111 u}$ are constants that need
not be zero. These conditions ensure the validity of the GIVs constructed in
the literature and considered in this section.\footnote{\cite
{gabaix2024granular} impose the stronger restrictions $\Greekmath 011B _{\Greekmath 0122
u}=0$ and $\Greekmath 011B _{\Greekmath 0111 u}=0$ for the identification of $\Greekmath 011E $ and $\Greekmath 0120 $
(see the first sentence of the paragraph containing their display (3)). As we show
below, these restrictions are not necessary. Moreover, as shown in the next
section, identification and estimation of $\Greekmath 011E $ and $\Greekmath 0120 $ do not require
the distributions of the aggregate and idiosyncratic shocks to be
time-invariant. In particular, their variances and covariances, such as $
\mathrm{Var}(u_{i,t})$ and $\mathrm{Cov}(\Greekmath 0111 _{t},u_{t})$, are allowed to
vary over time.}
Following \cite{gabaix2024granular}, we use the equally weighted average $
y_{e,t}\equiv e^{\top}y_{t}$, where $e\equiv n^{-1}\mathbf{1}_{n}$, to
construct a GIV defined as
\begin{equation}
z_{t}(e)\equiv y_{S,t}-y_{e,t}. \label{GIV_1}
\end{equation}
From the demand equation~(\ref{S_demand}), this GIV satisfies $z_{t}(e)=u_{t}^{\top}(S-e)$. Together with the first condition in (\ref{GIV_Validity}), this implies
\begin{equation}
\mathbb{E}[(p_{t}-\Greekmath 0120 y_{S,t})z_{t}(e)]=\mathbb{E}[\Greekmath 0122 _{t}u_{t}^{
\top}](S-e)=\Greekmath 011B _{\Greekmath 0122 u}\mathbf{1}_{n}^{\top}(S-e)=0.
\label{S_moment_1}
\end{equation}
Similarly,
\begin{align}
\mathbb{E}[(y_{e,t}-\Greekmath 011E p_{t})z_{t}(e)] & =\mathbb{E}[(
\Greekmath 0111 _{t}+u_{e,t})u_{t}^{\top}](S-e) \notag \\
& =\mathrm{Cov}(\Greekmath 0111 _{t},u_{t})(S-e)+e^{\top}\mathrm{Var}(u_{t})(S-e)
\notag \\
& =\Greekmath 011B _{\Greekmath 0111 u}\mathbf{1}_{n}^{\top}(S-e)+\Greekmath 011B _{u}^{2}e^{\top}(S-e)=0.
\label{S_moment_2}
\end{align}
The GIV $z_{t}(e)$ thus provides moment conditions (\ref{S_moment_1}) and (
\ref{S_moment_2}) for identifying and estimating the elasticities $\Greekmath 011E $ and
$\Greekmath 0120 $.
The above identification strategy can be generalized to construct generic
GIVs
\begin{equation}
z_{t}(a)\equiv y_{S,t}-y_{a,t}=(S-a)^{\top}u_{t},
\label{GIV_Simple_General}
\end{equation}
where $a\in \mathbb{R}^{n}$ satisfies
\begin{equation}
a\neq S\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\qquad a^{\top}\mathbf{1}_{n}=1.
\label{IV_validity_rest}
\end{equation}
Under these conditions, moment restrictions analogous to (\ref{S_moment_1})
and (\ref{S_moment_2}) can be constructed:
\begin{align}
\mathbb{E}[(p_{t}-\Greekmath 0120 y_{S,t})z_{t}(a)] & =0, \label{S_moment_3} \\
\mathbb{E}[(y_{e,t}-\Greekmath 011E p_{t})z_{t}(a)] & =0. \label{S_moment_4}
\end{align}
The GIV in (\ref{GIV_1}) corresponds to the special case $a=e$.
When $n>2$, there exist multiple vectors $a$ satisfying (\ref
{IV_validity_rest}). Therefore multiple GIVs are available and $\Greekmath 0120 $ and $
\Greekmath 011E $ become over-identified. This provides a natural motivation for using
multiple GIVs both to improve efficiency and to conduct specification tests
of instrument validity. We next characterize the resulting set of moment
conditions and clarify its connection to the GIV-based approach.
The restrictions in (\ref{GIV_Validity}) imply the following moment
conditions:
\begin{align}
\mathbb{E}[(y_{t}-\Greekmath 011E p_{t}\mathbf{1}_{n})(y_{t}-\Greekmath 011E p_{t}\mathbf{1}
_{n})^{\top}] & =(\mathbb{E}[\Greekmath 0111 _{t}^{2}]+2\Greekmath 011B _{\Greekmath 0111 u})\mathbf{1}_{n}
\mathbf{1}_{n}^{\top}+\Greekmath 011B _{u}^{2}\mathbf{I}_{n}, \label{eq_moment_1} \\
\mathbb{E}[(p_{t}-\Greekmath 0120 y_{S,t})(y_{t}-\Greekmath 011E p_{t}\mathbf{1}_{n})] & =(\mathbb{
E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u})\mathbf{1}_{n},
\label{eq_moment_2}
\end{align}
which together provide $n(n+3)/2$ moment conditions.\ To separate the
parameters of interest from the nuisance parameters, let $
Q\equiv(q_{1},\ldots,q_{n})$ be an $n\times n$ orthonormal matrix with
\begin{equation}
q_{1}\equiv n^{-1/2}\mathbf{1}_{n}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ \ and \ }q_{j}
\equiv(j(j-1))^{-1/2}\left( \sum_{l\leq j-1}\ell_{l}-(j-1)\ell_{j}\right)
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ for }j\geq2, \label{ortho_q_j}
\end{equation}
where $\ell_{j}$ denotes the $j$th canonical basis vector of $\mathbb{R}^{n}$
. The following lemma provides a non-redundant representation of these
moment conditions.
\begin{lemma}
\label{L1_eq_moment} The non-redundant restrictions in (\ref{eq_moment_1})
can be equivalently written as
\begin{align}
\mathbb{E}\!\left[ (y_{e,t}-\Greekmath 011E p_{t})Q_{-1}^{\top }y_{t}\right] & =\mathbf{
0}_{n-1}, \label{L1_eq_moment_d1} \\
\mathrm{vech}\!\left( \mathbb{E}[Q_{-1}^{\top }y_{t}y_{t}^{\top
}Q_{-1}]-\Greekmath 011B _{u}^{2}\mathbf{I}_{n-1}\right) & =\mathbf{0}_{n(n-1)/2},
\label{L1_eq_moment_d2} \\
\mathbb{E}[(y_{e,t}-\Greekmath 011E p_{t})^{2}]-(\mathbb{E}[\Greekmath 0111 _{t}^{2}]+2\Greekmath 011B
_{\Greekmath 0111 u})-n^{-1}\Greekmath 011B _{u}^{2}& =0, \label{L1_eq_moment_d3}
\end{align}
while the restrictions in (\ref{eq_moment_2}) can be equivalently written as
\begin{align}
\mathbb{E}\!\left[ (p_{t}-\Greekmath 0120 y_{S,t})Q_{-1}^{\top }y_{t}\right] & =\mathbf{
0}_{n-1}, \label{L1_eq_moment_s1} \\
\mathbb{E}\!\left[ (p_{t}-\Greekmath 0120 y_{S,t})(y_{e,t}-\Greekmath 011E p_{t})\right] -(\mathbb{
E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u})& =0,
\label{L1_eq_moment_s2}
\end{align}
where $Q_{-1}\equiv (q_{2},\ldots ,q_{n})$ and $\{{q_{j}\}}_{j=1}^{n}$ is
defined in (\ref{ortho_q_j}).
\end{lemma}
The moment conditions in (\ref{L1_eq_moment_d1}) and (\ref{L1_eq_moment_s1})
are equivalent to those in (\ref{S_moment_3}) and (\ref{S_moment_4}), and
can be directly used within a GMM framework for estimation and inference of
the unknown elasticities $\Greekmath 0120 $ and $\Greekmath 011E $.\footnote{
To establish this equivalence, note first that (\ref{L1_eq_moment_d1}) and (
\ref{L1_eq_moment_s1}) are constructed using GIVs of the form $q_{j}^{\top
}y_{t}$ for $j\geq2$. For any $a\in \mathbb{R}^{n}$ satisfying (\ref
{IV_validity_rest}), the corresponding GIV is $(S-a)^{\top}y_{t}$, where $
(S-a)^{\top}\mathbf{1}_{n}=0$. Since $Q_{-1}$ spans the subspace orthogonal
to $\mathbf{1}_{n}$, it follows that $S-a$ can be written as a linear
combination of the columns of $Q_{-1}$. Hence, the moment conditions in (\ref
{S_moment_3}) and (\ref{S_moment_4}) are implied by those in (\ref
{L1_eq_moment_d1}) and (\ref{L1_eq_moment_s1}). Conversely, for each $j\geq2$
, the vector $S-q_{j}$ satisfies (\ref{IV_validity_rest}), implying that (
\ref{L1_eq_moment_d1}) and (\ref{L1_eq_moment_s1}) are implied by (\ref
{S_moment_3}) and (\ref{S_moment_4}).} In contrast, the restrictions in (\ref
{L1_eq_moment_d2})--(\ref{L1_eq_moment_d3}) and (\ref{L1_eq_moment_s2})
involve only nuisance parameters, namely $\Greekmath 011B _{u}^{2}$, $\mathbb{E}
[\Greekmath 0111 _{t}^{2}]+2\Greekmath 011B _{\Greekmath 0111 u}$, and $\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111
_{t}]+\Greekmath 011B _{\Greekmath 0122 u}$.
Specifically, the moment conditions in (\ref{L1_eq_moment_d2}) identify $
\Greekmath 011B _{u}^{2}$ and also yield additional restrictions that do not depend on
unknown parameters. Conditional on $\Greekmath 011E $, $\Greekmath 0120 $, and $\Greekmath 011B _{u}^{2}$, the
quantities $\mathbb{E}[\Greekmath 0111 _{t}^{2}]+2\Greekmath 011B _{\Greekmath 0111 u}$ and $\mathbb{E}
[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u}$ are just-identified by (
\ref{L1_eq_moment_d3}) and (\ref{L1_eq_moment_s2}), respectively. Since $
\Greekmath 011B _{u}^{2}$ is over-identified by (\ref{L1_eq_moment_d2}), jointly
estimating $\Greekmath 011E $, $\Greekmath 0120 $, and $\Greekmath 011B _{u}^{2}$ using (\ref{L1_eq_moment_d1}
), (\ref{L1_eq_moment_d2}), and (\ref{L1_eq_moment_s1}) may yield more
efficient estimators of $\Greekmath 011E $ and $\Greekmath 0120 $ than those based only on (\ref
{L1_eq_moment_d1}) and (\ref{L1_eq_moment_s1}); see, for example, \cite
{ackerberg2014asymptotic}.
The analysis in this section relies on a simplified demand specification in
which the aggregate shock $\Greekmath 0111 _{t}$ enters with a known and homogeneous
loading across entities. In many applications, however, aggregate shocks may
have heterogeneous effects that are not directly observed, giving rise to a
more general factor structure. In the next section, we extend the GIV
framework to this setting, where the demand equation includes unobserved
factors with unknown loadings. This introduces new identification and
estimation challenges, as the moment conditions derived above are no longer
directly applicable when the factor loadings are unknown.
\section{Granular IVs in a General Model \label{sec: G_model}}
In this section, we study estimation and inference using GIVs in a more
general model in which the demand equation (\ref{S_demand}) incorporates a
set of unobserved factors with unknown factor loadings.\footnote{
The model (\ref{G_demand})-(\ref{G_supply}) can be further extended to
include exogenous regressors in both the demand and supply equations without
affecting the nature of the estimation and inference procedures proposed in
this section; see Subsection \ref{subsec: Ex1} for details.}\ Specifically,
we consider
\begin{align}
y_{t} & =\Greekmath 011E p_{t}\mathbf{1}_{n}+\Greekmath 0115 \Greekmath 0111 _{t}+u_{t}, \label{G_demand}
\\
p_{t} & =\Greekmath 0120 y_{S,t}+\Greekmath 0122 _{t}. \label{G_supply}
\end{align}
Here $\Greekmath 0111 _{t}$ denotes an $r\times1$ vector of unobserved factors, and $
\Greekmath 0115 $ is an $n\times r$ matrix of factor loadings. The vector $\Greekmath 0111 _{t}$
may also include a constant term, in which case the corresponding loading
captures unobserved entity fixed effects. While the supply equation appears
identical to (\ref{S_supply}),\ we now define\
\begin{equation*}
y_{S,t}=S_{t}^{\top}y_{t},
\end{equation*}
where $S_{t}\equiv(s_{i,t})_{i\leq n}$, and $s_{i,t}$ is nonnegative and
predetermined at time $t$.\footnote{
Throughout this section, we assume that the number of entities $n$ is fixed
over $t$. The identification strategy, as well as the estimation and
inference procedures proposed in this section, also apply to settings in
which $n$ varies over time; see Subsection \ref{subsec: Ex2} for details.}
In contrast to the simplified model studied in the previous section, we now
allow the factor loadings $\Greekmath 0115 $ to be unknown, which renders the earlier
identification and estimation results inapplicable and constitutes the main
challenge addressed in this section. One approach to handling the unobserved
factors $\Greekmath 0111 _{t}$ is to estimate them from $y_{t}$ (after partialling out $
p_{t}$ and entity fixed effects) using principal component analysis (see,
e.g., \cite{gabaix2021search} and \cite{banafti2022inferential}). However,
as noted in the literature (see, e.g., \cite{bai2003inferential}), the
consistency of the estimated factors typically requires the number of
entities $n$ to diverge, which stands in sharp contrast to most applications
of GIVs, where the number of entities is relatively small.
Another approach, proposed in \cite{gabaix2024granular}, attempts to
identify $\Greekmath 0115 $ jointly with the other unknown parameters in the model
under normalization restrictions and conditions similar to (but stronger
than) Assumption \ref{ID} below. However, as we show in Subsection \ref
{subsec: GK}, their identification strategy fails to identify the factor
loadings and may lead to invalid inference.
The method proposed in this section is based on an identification result
that applies for any $n$, whether finite or diverging. Although we follow
\cite{gabaix2024granular} and construct our inference procedures under an
asymptotic framework with fixed $n$, as discussed in Subsection \ref{subsec:
Ex2}, the method can be straightforwardly extended to settings in which $n$
is large or even exceeds $T$.
\subsection{Identification \label{subsec: G_model_ID}}
In this subsection, we first establish identification of the demand and
supply elasticities $\Greekmath 011E $ and $\Greekmath 0120 $ given $\Greekmath 0115 $, thereby extending the
results in Lemma \ref{L1_eq_moment}. We then provide a constructive identification result for the orthogonal complement of $\func{col}((\mathbf{1}_{n}, \Greekmath 0115 ))$, which forms the basis for the
estimation and inference procedures developed in the next subsection. We
begin by stating the conditions required for identification.
\begin{assumption}
\label{ID} (i) $\mathbb{E}[u_{t}]=\mathbf{0}_{n}$ and $\mathrm{Var}
(u_{t})=\Greekmath 011B _{u,t}^{2}\mathbf{I}_{n}$; (ii) $\mathrm{Cov}
(\Greekmath 0111 _{t},u_{t})=\Gamma_{\Greekmath 0111 u,t}\mathbf{1}_{n}^{\top}$, where $
\Gamma_{\Greekmath 0111 u,t}$ is an $r\times1$ vector; (iii) $\mathrm{Cov}
(\Greekmath 0122 _{t},u_{t})=\Greekmath 011B _{\Greekmath 0122 u,t}\mathbf{1}_{n}^{\top}$,
where $\Greekmath 011B _{\Greekmath 0122 u,t}$ is a finite scalar; (iv) $T^{-1}\sum_{t\leq
T}\mathbb{E}[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top }]$ is nonsingular and $n>\bar{r}$,
where $\bar{r}\equiv \mathrm{rank}((\mathbf{1}_{n},\Greekmath 0115 ))$.
\end{assumption}
Assumption \ref{ID}(i)--(iii) generalize the conditions in (\ref
{GIV_Validity}) by allowing the joint distribution of the demand shocks $
u_{t}$, the supply shocks $\Greekmath 0122 _{t}$, and the factors $\Greekmath 0111 _{t}$ to
vary over time. Under these conditions, we obtain
\begin{align}
\mathbb{E}[(y_{t}-\Greekmath 011E p_{t}\mathbf{1}_{n})(y_{t}-\Greekmath 011E p_{t}\mathbf{1}
_{n})^{\top}] & =\Greekmath 0115 \mathbb{E}[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]\Greekmath 0115 ^{\top
}+\Greekmath 0115 \Gamma_{\Greekmath 0111 u,t}\mathbf{1}_{n}^{\top}+\mathbf{1}_{n}\Gamma_{\Greekmath 0111
u,t}^{\top}\Greekmath 0115 ^{\top}+\Greekmath 011B _{u,t}^{2}\mathbf{I}_{n}, \label{G_moment_1}
\\
\mathbb{E}[(p_{t}-\Greekmath 0120 y_{S,t})(y_{t}-\Greekmath 011E p_{t}\mathbf{1}_{n})] & =\Greekmath 0115
\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u,t}\mathbf{1}_{n},
\label{G_moment_2}
\end{align}
which provide a total of $n(n+3)/2$ moment conditions for the unknown
parameters $\Greekmath 011E $, $\Greekmath 0120 $, $\Greekmath 011B _{u,t}^{2}$, $\mathbb{E}[\Greekmath 0111 _{t}\Greekmath 0111
_{t}^{\top}]$, $\Gamma_{\Greekmath 0111 u,t}$, $\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]$,
and $\Greekmath 011B _{\Greekmath 0122 u,t}$. Assumption \ref{ID}(iv) is primarily imposed
to ensure identification of $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$, whose
orthogonal complement will be used to construct the GIVs.
We now reorganize the moment conditions (\ref{G_moment_1})-(\ref{G_moment_2}
) according to their roles in identifying the different parameters.
\begin{lemma}
\label{L2_eq_moment}\ Let $\bar{\Greekmath 0115 }\equiv(n^{-1/2}\mathbf{1}_{n},\bar{
\Greekmath 0115 }_{-1})$ be an $n\times \bar{r}$ orthonormal matrix spanning $\func{col}((\mathbf{1}_{n}, \Greekmath 0115 ))$, and let $\bar{\Greekmath 0115 }_{\bot}$ denote its
orthonormal complement. Then the non-redundant restrictions in (\ref
{G_moment_1}) can be equivalently expressed as
\begin{align}
\mathbb{E}\! \left[ (y_{e,t}-\Greekmath 011E p_{t})\bar{\Greekmath 0115 }_{\bot}^{\top}y_{t}
\right] & =\mathbf{0}_{n-\bar{r}}, \label{L2_eq_moment_1} \\
\mathbb{E}\! \left[ \bar{\Greekmath 0115 }_{-1}^{\top}y_{t}y_{t}^{\top}\bar{\Greekmath 0115 }
_{\bot}\right] & =\mathbf{0}_{(\bar{r}-1)\times(n-\bar{r})},
\label{L2_eq_moment_2} \\
\mathrm{vech}\! \left( \mathbb{E}\! \left[ \bar{\Greekmath 0115 }_{\bot}^{
\top}y_{t}y_{t}^{\top}\bar{\Greekmath 0115 }_{\bot}\right] -\Greekmath 011B _{u,t}^{2}\mathbf{I}
_{n-\bar{r}}\right) & =\mathbf{0}_{(n-\bar{r}+1)(n-\bar{r})/2},
\label{L2_eq_moment_3}
\end{align}
and
\begin{align}
& \mathrm{vech}\! \left( \mathbb{E}\! \left[ \bar{\Greekmath 0115 }^{\top}(y_{t}-\Greekmath 011E
p_{t}\mathbf{1}_{n})(y_{t}-\Greekmath 011E p_{t}\mathbf{1}_{n})^{\top}\bar{\Greekmath 0115 }
\right] \right) \notag \\
& \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ \ \ \ \ \ \ \ \ \ }\overset{}{=}\mathrm{vech}\! \left( \bar{
\Greekmath 0115 }^{\top}\Big(\Greekmath 0115 \mathbb{E}[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]\Greekmath 0115 ^{
\top}+\Greekmath 0115 \Gamma_{\Greekmath 0111 u,t}\mathbf{1}_{n}^{\top}+\mathbf{1}
_{n}\Gamma_{\Greekmath 0111 u,t}^{\top}\Greekmath 0115 ^{\top}+\Greekmath 011B _{u,t}^{2}\mathbf{I}_{n}
\Big)\bar{\Greekmath 0115 }\right) , \label{L2_eq_moment_4}
\end{align}
while the restrictions in (\ref{G_moment_2}) can be equivalently written as
\begin{align}
\mathbb{E}\! \left[ (p_{t}-\Greekmath 0120 y_{S,t})\bar{\Greekmath 0115 }_{\bot}^{\top}y_{t}
\right] & =\mathbf{0}_{n-\bar{r}}, \label{L2_eq_moment_5} \\
\mathbb{E}\! \left[ \bar{\Greekmath 0115 }^{\top}(p_{t}-\Greekmath 0120 y_{S,t})(y_{t}-\Greekmath 011E p_{t}
\mathbf{1}_{n})\right] -\big(\bar{\Greekmath 0115 }^{\top}\Greekmath 0115 \mathbb{E}
[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]+\Greekmath 011B _{\Greekmath 0122 u,t}\bar{\Greekmath 0115 }^{\top }
\mathbf{1}_{n}\big) & =\mathbf{0}_{\bar{r}}. \label{L2_eq_moment_6}
\end{align}
\end{lemma}
Lemma \ref{L2_eq_moment} shows that the key moment conditions for
identifying $\Greekmath 011E $ and $\Greekmath 0120 $ are given by (\ref{L2_eq_moment_1}) and (\ref
{L2_eq_moment_5}), which are constructed from the generalized GIVs $\bar{
\Greekmath 0115 }_{\bot}^{\top}y_{t}$. The moment conditions associated with the
diagonal elements of
\begin{equation*}
\mathbb{E}\! \left[ \bar{\Greekmath 0115 }_{\bot}^{\top}y_{t}y_{t}^{\top}\bar{\Greekmath 0115
}_{\bot}\right] -\Greekmath 011B _{u,t}^{2}\mathbf{I}_{n-\bar{r}}=\mathbf{0}_{(n-\bar{r
})\times(n-\bar{r})}
\end{equation*}
provide identifying restrictions for $T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}$.
In contrast, the moment conditions in (\ref{L2_eq_moment_2}), as well as the
off-diagonal elements of the matrix above, do not involve any unknown
parameters and are therefore redundant from an identification standpoint.
Nevertheless, they may be useful for improving the efficiency of the GMM
estimator and for testing specification assumptions, such as Assumption \ref
{ID}(i). Finally, (\ref{L2_eq_moment_4}) and (\ref{L2_eq_moment_6}) impose
restrictions on the nuisance parameters $T^{-1}\sum_{t\leq T}\mathbb{E}
[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]$, $T^{-1}\sum_{t\leq T}\Gamma_{\Greekmath 0111 u,t}$, $
T^{-1}\sum_{t\leq T}\mathbb{E}[\Greekmath 0122 _{t}\Greekmath 0111 _{t}]$, and $
T^{-1}\sum_{t\leq T}\Greekmath 011B _{\Greekmath 0122 u,t}$, conditional on the
identification of $\Greekmath 011E $, $\Greekmath 0120 $, and $T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}$.
The moment conditions in (\ref{L2_eq_moment_1}) and (\ref{L2_eq_moment_5})
for the identification of $\Greekmath 011E $ and $\Greekmath 0120 $ rely on the generalized GIVs $
\bar{\Greekmath 0115 }_{\bot}^{\top}y_{t}$. These instruments are, however,
infeasible in practice when the factor loading matrix $\Greekmath 0115 $ is unknown.
We therefore next show how to identify the column space of $\bar{\Greekmath 0115 }$,
which in turn determines the space spanned by $\bar{\Greekmath 0115 }_{\bot}$.
To this end, let $M_{\mathbf{1}_{n}}\equiv \mathbf{I}_{n}-n^{-1}\mathbf{1}
_{n}\mathbf{1}_{n}^{\top}$ and define the demeaned variables
\begin{equation}
\tilde{y}_{t}\equiv M_{\mathbf{1}_{n}}y_{t},\qquad \tilde{\Greekmath 0115 }\equiv M_{
\mathbf{1}_{n}}\Greekmath 0115 ,\qquad \tilde{u}_{t}\equiv M_{\mathbf{1}_{n}}u_{t}.
\label{tilda_vars}
\end{equation}
Applying $M_{\mathbf{1}_{n}}$ to both sides of (\ref{G_demand}) yields
\begin{equation}
\tilde{y}_{t}=\tilde{\Greekmath 0115 }\Greekmath 0111 _{t}+\tilde{u}_{t}.
\label{Demeaned_Demand}
\end{equation}
Combining (\ref{Demeaned_Demand}) with Assumption \ref{ID}, we obtain
\begin{equation}
\mathbb{E}[\tilde{y}_{t}\tilde{y}_{t}^{\top}]=\tilde{\Greekmath 0115 }\mathbb{E}
[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]\tilde{\Greekmath 0115 }^{\top}+\Greekmath 011B _{u,t}^{2}M_{\mathbf{1}
_{n}}. \label{G_demand_factor_1}
\end{equation}
Averaging (\ref{G_demand_factor_1}) over $t$ then yields
\begin{equation}
\bar{\Sigma}_{\tilde{y}}=\tilde{\Greekmath 0115 }\left( T^{-1}\sum_{t\leq T}\mathbb{E}
[\Greekmath 0111 _{t}\Greekmath 0111 _{t}^{\top}]\right) \tilde{\Greekmath 0115 }^{\top}+\bar {\Greekmath 011B }
_{u}^{2}M_{\mathbf{1}_{n}} \label{G_demand_factor}
\end{equation}
where
\begin{equation*}
\bar{\Sigma}_{\tilde{y}}\equiv T^{-1}\sum_{t\leq T}\mathbb{E}[\tilde{y}_{t}
\tilde{y}_{t}^{\top}]\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ }\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and \ }\qquad \bar{\Greekmath 011B }
_{u}^{2}\equiv T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}.
\end{equation*}
Here the matrix $\bar{\Sigma}_{\tilde{y}}$ is identified and can be
consistently estimated.\ The following lemma shows that given the
identification of $\bar{\Sigma}_{\tilde{y}}$, (\ref{G_demand_factor}) is
sufficient to identify both $\bar{\Greekmath 011B }_{u}^{2}$ and the subspace
orthogonal to the column space of $\bar{\Greekmath 0115 }$.
\begin{lemma}
\label{ID_G_GIV_Weight} Under Assumption \ref{ID}(iv),
\begin{equation}
\bar{\Greekmath 011B }_{u}^{2}=\min_{a\in \mathcal{B}_{\mathbf{1}_{n}}}a^{\top}\bar{
\Sigma}_{\tilde{y}}a, \label{G_demand_sigma_u}
\end{equation}
where
\begin{equation*}
\mathcal{B}_{\mathbf{1}_{n}}\equiv \left \{ a\in \mathbb{R}^{n}:\mathbf{1}
_{n}^{\top}a=0,\ \Vert a\Vert=1\right \} .
\end{equation*}
Moreover, the set of minimizers of (\ref{G_demand_sigma_u}) spans $\func{col}(\bar{
\Greekmath 0115 }_{\bot})$.
\end{lemma}
Lemma \ref{ID_G_GIV_Weight} provides a constructive characterization of $
\bar{\Greekmath 0115 }_{\bot}$, which is central to the construction of generalized
GIVs. In particular, $\bar{\Greekmath 0115 }_{\bot}$ can be recovered as the
eigenspace associated with the smallest eigenvalue of $\bar{\Sigma}_{\tilde{y
}}$ restricted to $\mathcal{B}_{\mathbf{1}_{n}}$. Intuitively, this
corresponds to extracting directions of cross-sectional variation in $y_{t}$
that are orthogonal to both the common factor structure and the aggregate
component spanned by $\mathbf{1}_{n}$.
The minimization problem in (\ref{G_demand_sigma_u}), however, is defined
over the constrained set $\mathcal{B}_{\mathbf{1}_{n}}$, which is not
directly convenient for implementation. To facilitate computation, we next
provide an equivalent representation that transforms this constrained
problem into an unconstrained eigenvalue problem in $\mathbb{R}^{n-1}$.
\begin{lemma}
\label{G_GIV_Solutions} Suppose Assumption \ref{ID}(iv) holds. Consider the
minimization problem:
\begin{equation}
\min_{\tilde{a}\in B_{n-1}}\tilde{a}^{\top}Q_{-1}^{\top}\bar{\Sigma}
_{y}Q_{-1}\tilde{a}, \label{G_GIV_Weights}
\end{equation}
where
\begin{equation*}
\bar{\Sigma}_{y}\equiv T^{-1}\sum_{t\leq T}\mathbb{E}[y_{t}y_{t}^{\top
}],\qquad B_{n-1}\equiv \{ \tilde{a}\in \mathbb{R}^{n-1}:\Vert \tilde{a}
\Vert=1\}.
\end{equation*}
Then $a$ is a minimizer of (\ref{G_demand_sigma_u}) if and only if there
exists a minimizer $\tilde{a}$ of (\ref{G_GIV_Weights}) such that $a=Q_{-1}
\tilde{a}$.
\end{lemma}
The solutions to (\ref{G_GIV_Weights}) are given by the normalized
eigenvectors associated with the smallest eigenvalue of the symmetric matrix
$Q_{-1}^{\top}\bar{\Sigma}_{y}Q_{-1}$. By Lemma \ref{ID_G_GIV_Weight} and
Lemma \ref{G_GIV_Solutions}, these eigenvectors, after left multiplication
by $Q_{-1}$, span the same subspace as $\bar{\Greekmath 0115 }_{\bot}$, thereby
providing a feasible representation of the generalized GIVs. In practice, $
Q_{-1}^{\top }\bar{\Sigma}_{y}Q_{-1}$ can be consistently estimated by its
sample analogue, so the GIVs can be implemented via standard eigenvalue
decomposition without requiring knowledge of the factor loadings $\Greekmath 0115 $.
\subsection{Estimation and inference with GIVs \label{subsec: G_model_EI}}
Building on Lemmas \ref{L2_eq_moment}--\ref{G_GIV_Solutions} in the previous
subsection, the moment conditions (\ref{L2_eq_moment_1}) and (\ref
{L2_eq_moment_5}) for the identification and estimation of $\Greekmath 0112
\equiv(\Greekmath 011E ,\Greekmath 0120 )^{\top}$\ can now be written as
\begin{equation}
\mathbb{E}[\bar{g}_{T}(\Greekmath 0112 ;A)]=\mathbf{0}_{2(n-\bar{r})},
\label{Moment_Cond}
\end{equation}
where
\begin{equation}
\bar{g}_{T}(\Greekmath 0112 ;A)\equiv T^{-1}\sum_{t\leq T}
\begin{pmatrix}
A^{\top}y_{t}(y_{e,t}-\Greekmath 011E p_{t}) \\
A^{\top}y_{t}(p_{t}-\Greekmath 0120 y_{S,t})
\end{pmatrix}
. \label{Moment_Func}
\end{equation}
Here $A\equiv Q_{-1}A_{0}$, where $Q_{-1}$ is an $n\times(n-1)$ matrix
defined in Lemma \ref{L1_eq_moment}, and $A_{0}$ is an $(n-1)\times(n-\bar{r}
)$ matrix collecting the eigenvectors corresponding to the smallest $n-\bar{r
}$ eigenvalues of
\begin{equation*}
S_{y}\equiv Q_{-1}^{\top}\bar{\Sigma}_{y}Q_{-1}.
\end{equation*}
Since\ $A_{0}$ depends on the unknown population covariance matrix, the
moment function $\bar{g}_{T}(\Greekmath 0112 ;A)$ is not directly feasible in practice.
To construct feasible moment conditions, we replace $A$ in (\ref{Moment_Func}
) with $\hat{A}\equiv Q_{-1}\hat{A}_{0}$, where $\hat{A}_{0}$ collects the
eigenvectors corresponding to the smallest $n-\bar{r}$ eigenvalues of
\begin{equation*}
\hat{S}_{y}\equiv Q_{-1}^{\top}\hat{\Sigma}_{y}Q_{-1},\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{where}\
\hat{\Sigma}_{y}\equiv T^{-1}\sum_{t\leq T}y_{t}y_{t}^{\top}.
\end{equation*}
The GIV estimator is then defined as
\begin{equation}
\hat{\Greekmath 0112 }(\hat{A})\equiv \arg \min_{\Greekmath 0112 \in \Theta}\bar{g}_{T}(\Greekmath 0112 ;
\hat{A})^{\top}W_{0,T}(\hat{A})\bar{g}_{T}(\Greekmath 0112 ;\hat{A}),
\label{GMM_Criterion}
\end{equation}
where
\begin{equation}
W_{0,T}(\hat{A})\equiv((\mathbf{I}_{2}\otimes \hat{A}^{\top})W_{0,T}(\mathbf{
I}_{2}\otimes \hat{A}))^{-1}, \label{Weight}
\end{equation}
and $W_{0,T}$ is a user-specified symmetric positive definite $2n\times2n$
matrix.
Since $\bar{g}_{T}(\Greekmath 0112 ;\hat{A})$ is linear in $\Greekmath 0112 $, the GIV estimator
admits the closed-form representation
\begin{equation}
\hat{\Greekmath 0112 }(\hat{A})=\big(D_{1,T}(\hat{A})^{\top}W_{0,T}(\hat{A})D_{1,T}(
\hat{A})\big)^{-1}\big(D_{1,T}(\hat{A})^{\top}W_{0,T}(\hat{A})D_{2,T}(\hat {A
})\big), \label{GIV_Form_1}
\end{equation}
where $D_{j,T}(\hat{A})\equiv(\mathbf{I}_{2}\otimes \hat{A}^{\top})D_{j,T}$
for $j=1,2$, and
\begin{equation}
D_{1,T}\equiv T^{-1}\sum_{t\leq T}\mathrm{diag}\! \left(
y_{t}p_{t},\,y_{t}y_{S,t}\right) ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }\ D_{2,T}\equiv T^{-1}\sum_{t\leq
T}
\begin{pmatrix}
y_{t}y_{e,t} \\
y_{t}p_{t}
\end{pmatrix}
. \label{GIV_Form_2}
\end{equation}
We next present sufficient conditions for establishing the asymptotic
properties of $\hat{\Greekmath 0112 }(\hat{A})$ within the same framework as \cite
{gabaix2024granular}, where the number of entities $n$ is fixed and the
number of observations $T$ (indexed by $t$) tends to infinity.\footnote{
The asymptotic properties of $\hat{\Greekmath 0112 }(\hat{A})$, as well as inference
for the unknown parameter $\Greekmath 0112 $, can be extended to the case where both $n
$ and $T$ diverge by applying techniques from the many-moments literature;
see, for example, \cite{han2006gmm} and \cite{newey2009generalized}.} Let $
\{ \Greekmath 0116 _{j}\}_{j\leq n-1}$ denote the eigenvalues of $S_{y}$ arranged in
increasing order, and let $A_{0,\bot}$ denote the matrix collecting the
eigenvectors associated with $\{ \Greekmath 0116 _{j}\}_{n-\bar{r}+1\leq j\leq n-1}$.
\begin{assumption}
\label{Asy_Cond_1} (i) $\left \vert \Greekmath 011E \Greekmath 0120 -1\right \vert \geq K^{-1}$ and
$\left \vert \Greekmath 011E \right \vert +\left \vert \Greekmath 0120 \right \vert +\left \Vert
\Greekmath 0115 \right \Vert \leq K$;\ (ii)\ for $a,b\in \{u,\Greekmath 0111 ,\Greekmath 0122 \}$,
\begin{equation*}
T^{-1/2}\sum_{t\leq T}\big(a_{t}b_{t}^{\top}-\mathbb{E}[a_{t}b_{t}^{\top }]
\big)=O_{p}(1);
\end{equation*}
(iii) $\Greekmath 0116 _{n-\bar{r}+1}-\bar{\Greekmath 011B }_{u}^{2}>K^{-1}$ and\ $\bar{\Greekmath 011B }
_{u}^{2}>K^{-1}$;\ (iv) $\max_{t\leq T}\mathbb{E}[u_{t}^{\top}u_{t}+
\Greekmath 0122 _{t}^{2}+\Greekmath 0111 _{t}^{\top}\Greekmath 0111 _{t}]\leq K$.
\end{assumption}
Assumption \ref{Asy_Cond_1}(i) ensures that the demand and supply system
admits a well-defined reduced form and, consequently, a unique equilibrium.
Assumption \ref{Asy_Cond_1}(ii) guarantees that the population second
moments of $(u_{t},\Greekmath 0122 _{t},\Greekmath 0111 _{t})$ are approximated by their
sample counterparts at the rate $T^{-1/2}$. From Lemma \ref{ID_G_GIV_Weight}
, we have $\Greekmath 0116 _{j}=T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}$ for all $j\leq n-
\bar{r}$. Therefore, Assumption \ref{Asy_Cond_1}(iii) imposes an eigenvalue
gap condition on $S_{y}$, which is essential for consistent estimation of
the eigenspace $\bar{\Greekmath 0115 }_{\bot}$. This condition can be verified under
a lower bound condition on $\Greekmath 011A _{\min}((\mathbf{1}_{n},\Greekmath 0115 )^{\top }(
\mathbf{1}_{n},\Greekmath 0115 ))$, or on $\Greekmath 011A _{\min}(\Greekmath 0115 ^{\top}\Greekmath 0115 )$ when $
\mathbf{1}_{n}\in \func{col}(\Greekmath 0115 )$; see Lemma \ref{Suff_Cond1_iii} in
Online Appendix \ref{APP_3} for details. Assumption \ref{Asy_Cond_1}(iii)
also requires that the variance of the idiosyncratic demand shock be bounded
away from zero, which is important for maintaining sufficient identification
strength of the GIVs. Finally, Assumption \ref{Asy_Cond_1}(iv), together
with \ref{Asy_Cond_1}(i), ensures that the second moments of $y_{t}$ and $
p_{t}$ are well defined.
\begin{assumption}
\label{S} The sequence of market shares $\left \{ S_{t}\right \} $
satisfies: (i)
\begin{equation*}
T^{-1/2}\sum_{t\leq T}(a_{t}b_{t}^{\top}-{\mathbb{E}[}a_{t}b_{t}^{
\top}])=O_{p}(1)
\end{equation*}
for $a_{t},b_{t}\in \{S_{t}^{\top}u_{t},\Greekmath 0111 _{t}\otimes S_{t},\Greekmath 0122
_{t}\}$, or $a_{t}\in \{u_{t},\Greekmath 0111 _{t}\}$\ and\ $b_{t}\in
\{S_{t}^{\top}u_{t},\Greekmath 0111 _{t}\otimes S_{t}\}$; (ii)\ $\mathbf{1}
_{n}^{\top}S_{t}=1$ for all $t$.
\end{assumption}
Assumption \ref{S}(i) imposes a set of high-level moment conditions ensuring
that sample averages involving the weighted aggregates, such as $S_{t}^{\top
}u_{t}$, satisfy a standard $T^{-1/2}$ law of large numbers. In particular,
it requires that interactions between market-share weights and the
structural shocks $u_{t}$, $\Greekmath 0111 _{t}$ and $\Greekmath 0122 _{t}$\ exhibit
sufficiently weak temporal dependence and possess finite second moments.
This condition is analogous to Assumption \ref{Asy_Cond_1}(ii), and is
implied by it when $S_{t}$ is time-invariant. More generally, both
conditions can be verified under standard mixing or martingale difference
assumptions. Assumption \ref{S}(ii) is a normalization condition requiring
that the elements of $S_{t}$ sum to one.
Let $v_{t}\equiv y_{e,t}-\Greekmath 011E p_{t}$. For $b\in \{v,\Greekmath 0122 \}$, define
\begin{equation}
\Greekmath 0118 _{b,t}\equiv y_{t}b_{t}-\mathbb{E}[y_{t}b_{t}]+(y_{t}y_{t}^{\top }-
\mathbb{E}[y_{t}y_{t}^{\top}])\, \Upsilon \left( T^{-1}\sum_{t\leq T}\mathbb{
E}[y_{t}b_{t}]\right) , \label{zeta_b}
\end{equation}
where
\begin{equation*}
\Upsilon \equiv Q_{-1}A_{0,\bot}(\bar{\Greekmath 011B }_{u}^{2}\mathbf{I}_{\bar{r}
-1}-\Lambda_{\bot})^{-1}A_{0,\bot}^{\top}Q_{-1}^{\top},\qquad \Lambda_{\bot
}\equiv \mathrm{diag}((\Greekmath 0116 _{j})_{n-\bar{r}+1\leq j\leq n-1}).
\end{equation*}
The random vectors $\Greekmath 0118 _{b,t}$, for $b\in \{v,\Greekmath 0122 \}$, represent the
estimation errors in the moment conditions used to estimate $\Greekmath 011E $ and $\Greekmath 0120 $
, respectively. The first component, $y_{t}b_{t}-\mathbb{E}[y_{t}b_{t}]$,
captures the sampling variation that would arise even if the\ factor loading
matrix $\Greekmath 0115 $ were known. The second component reflects the additional
estimation error induced by replacing $\bar{\Greekmath 0115 }_{\bot}$ with its
estimator, and hence accounts for the impact of estimating the loading
matrix on the moment conditions.
\begin{assumption}
\label{Asy_Cond_2} (i) $V^{-1/2}T^{-1/2}\sum_{t\leq T}\Greekmath 0118 _{t}\rightarrow
_{d}N(0,\mathbf{I}_{2n})$ where $\Greekmath 0118 _{t}\equiv(\Greekmath 0118 _{v,t}^{\top},\Greekmath 0118
_{\Greekmath 0122 ,t}^{\top})^{\top}$ with $\Greekmath 011A _{\min}(V)\geq K^{-1}$; (ii) $
W_{0,T}=W_{0}+o_{p}(1)$, where $W_{0}$ is a nonrandom symmetric matrix with $
K^{-1}\leq \Greekmath 011A _{\min}(W_{0})\leq \Greekmath 011A _{\max}(W_{0})\leq K$; (iii) $\Vert
T^{-1}\sum_{t\leq T}A^{\top}\mathbb{E}[y_{t}p_{t}]\Vert \geq K^{-1}$ and $
\Vert T^{-1}\sum_{t\leq T}A^{\top}\mathbb{E}[y_{t}y_{S,t}]\Vert \geq K^{-1}$
; (iv) there exists a matrix {$\hat{V}$ such that $\hat{V}=V+o_{p}(1)$.}
\end{assumption}
Assumption \ref{Asy_Cond_2}(i) concerns the asymptotic distribution of $
V^{-1/2}T^{-1/2}\sum_{t\leq T}\Greekmath 0118 _{t}$, which can be established via a
central limit theorem. Here $V$ denotes the variance matrix of $
T^{-1/2}\sum_{t\leq T}\Greekmath 0118 _{t}$. Assumption \ref{Asy_Cond_2}(ii) ensures
consistent estimation of the weighting matrix, while Assumption \ref
{Asy_Cond_2}(iii) guarantees that the GIVs provide sufficient identification
strength for $\Greekmath 011E $ and $\Greekmath 0120 $ to be $T^{1/2}$-estimable.\ The latter
condition essentially requires that the weighted mean of the entities'
market shares,\ $S_{u}\equiv T^{-1}\sum_{t\leq T}\Greekmath 011B _{u,t}^{2}\mathbb{E}
[S_{t}]$ does not lie in $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$. It can be
verified under suitable primitive conditions; see Lemma \ref{Suff_Cond2_iii}
in Online Appendix \ref{APP_3} for details. Finally, Assumption \ref
{Asy_Cond_2}(iv) requires the existence of a consistent estimator of $V$.
\footnote{
A consistent estimator of $V$ can be constructed using the estimated shocks $
\hat{v}_{t}$ and $\hat{\Greekmath 0122 }_{t}$ obtained from the GIV estimator $
\hat{\Greekmath 0112 }(\hat{A})$ with identity weighting matrix $W_{0,T}=\mathbf{I}
_{2n}$; see, for example, (\ref{V_est}) in the implementation algorithm in
Online Appendix \ref{APP_0}. The consistency of this variance estimator is
established in Theorem \ref{V_Est} in the Online Appendix.}
To simplify the notation for the asymptotic variance of the GIV estimator,
define
\begin{equation*}
\Gamma(D_{1},W_{0},A)\equiv(D_{1}(A)^{\top}W_{0}(A)D_{1}(A))^{-1}D_{1}(A)^{
\top}W_{0}(A),
\end{equation*}
where $D_{1}(A)\equiv(\mathbf{I}_{2}\otimes A^{\top})D_{1},$
\begin{equation*}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }D_{1}\equiv T^{-1}\sum_{t\leq T}\mathbb{E}\! \left[ \mathrm{diag}\!
\left( y_{t}p_{t},\,y_{t}y_{S,t}\right) \right] \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ \ \ and \ \ \ }
W_{0}(A)\equiv((\mathbf{I}_{2}\otimes A^{\top})\,W_{0}\,(\mathbf{I}
_{2}\otimes A))^{-1}.
\end{equation*}
The following theorem establishes the asymptotic distribution of the GIV
estimator.
\begin{theorem}
\label{Asy_Dist} Under Assumptions \ref{ID}, \ref{Asy_Cond_1}, \ref{S}, and
\ref{Asy_Cond_2}(i)--(iii),
\begin{equation}
(\Gamma(D_{1},W_{0},A)V(A)\Gamma(D_{1},W_{0},A)^{\top})^{-1/2}T^{1/2}(\hat{
\Greekmath 0112 }(\hat{A})-\Greekmath 0112 )\; \rightarrow_{d}\;N(0,\mathbf{I}_{2}),
\label{Asy_Dist_1}
\end{equation}
where
\begin{equation*}
V(A)\equiv(\mathbf{I}_{2}\otimes A^{\top})\,V\,(\mathbf{I}_{2}\otimes A).
\end{equation*}
Moreover, if Assumption \ref{Asy_Cond_2}(iv) also holds, then
\begin{equation}
\Gamma(D_{1,T},W_{0,T},\hat{A})\hat{V}(\hat{A})\Gamma(D_{1,T},W_{0,T},\hat {A
})^{\top}=\Gamma(D_{1},W_{0},A)V(A)\Gamma(D_{1},W_{0},A)^{\top}+o_{p}(1),
\label{Asy_Dist_2}
\end{equation}
where
\begin{equation*}
\hat{V}(\hat{A})\equiv(\mathbf{I}_{2}\otimes \hat{A}^{\top})\, \hat {V}\,(
\mathbf{I}_{2}\otimes \hat{A}),
\end{equation*}
and $\Gamma(D_{1,T},W_{0,T},\hat{A})$ is defined analogously to $\Gamma
(D_{1},W_{0},A)$ with $D_{1}$, $W_{0}$, and $A$ replaced by $D_{1,T}$, $
W_{0,T}$, and $\hat{A}$, respectively.
\end{theorem}
Theorem \ref{Asy_Dist} shows that the asymptotic variance of the GIV
estimator is minimized when $W_{0}=V$. Accordingly, the optimal weighting
matrix can be obtained by setting $W_{0,T}=\hat{V}$ in (\ref{Weight}), which
yields
\begin{equation*}
W_{\ast,T}(\hat{A})\equiv \hat{V}(\hat{A})^{-1}.
\end{equation*}
Let $\hat{\Greekmath 0112 }^{\ast}(\hat{A})$ denote the corresponding optimally
weighted GIV estimator. It then follows that
\begin{equation}
(D_{1}(A)^{\top}V(A)^{-1}D_{1}(A))^{1/2}T^{1/2}\bigl(\hat{\Greekmath 0112 }^{\ast}(
\hat{A})-\Greekmath 0112 \bigr)\rightarrow_{d}N(0,\mathbf{I}_{2}).
\label{Asy_Dist_OGMM}
\end{equation}
Standard errors for $\hat{\Greekmath 0112 }^{\ast}(\hat{A})$ can be constructed from
the square roots of the diagonal elements of
\begin{equation}
\bigl(TD_{1,T}(\hat{A})^{\top}W_{\ast,T}(\hat{A})D_{1,T}(\hat{A})\bigr)
^{-1}, \label{GIV_Est_STD}
\end{equation}
whose validity follows from (\ref{Asy_Dist_2}).
Since there are $2(n-\bar{r})$ moment conditions for the identification and
estimation of $\Greekmath 011E $ and $\Greekmath 0120 $, these parameters are over-identified
whenever $n>\bar{r}+1$. In this case, the validity of the GIVs can be
assessed using an over-identification test.
\begin{theorem}
\label{J_Test} Under Assumptions {\ref{ID}, \ref{Asy_Cond_1}, \ref{S}, and
\ref{Asy_Cond_2}},
\begin{equation*}
T\bar{g}_{T}(\hat{\Greekmath 0112 }^{\ast}(\hat{A});\hat{A})^{\top}W_{\ast,T}(\hat {A})
\bar{g}_{T}(\hat{\Greekmath 0112 }^{\ast}(\hat{A});\hat{A})\rightarrow_{d}\Greekmath 011F ^{2}
\bigl(2(n-\bar{r}-1)\bigr).
\end{equation*}
\end{theorem}
Theorem \ref{J_Test} establishes the asymptotic distribution of the J-test
statistic under the null hypothesis that the moment conditions (\ref
{Moment_Cond}) are valid. When the GIVs are invalid, the power of the J-test
follows from standard GMM arguments and is therefore omitted for brevity.
\begin{remark}
Theorems \ref{Asy_Dist} and \ref{J_Test} establish estimation and inference
procedures for the demand and supply elasticities based on the full set of
moment conditions. In some applications, however, interest may center on a
single structural parameter, making it natural to estimate the demand and
supply elasticities separately using the corresponding subsets of moment
conditions.
For example, when the demand elasticity is the primary parameter of
interest, estimation may be based on the moment functions
\begin{equation}
\bar{g}_{\Greekmath 011E ,T}(\Greekmath 011E ;A)\equiv T^{-1}\sum_{t\leq
T}A^{\top}y_{t}(y_{e,t}-\Greekmath 011E p_{t}). \label{Moment_phi}
\end{equation}
Using arguments analogous to those in the proof of Theorem \ref{Asy_Dist},
the resulting GIV estimator can be shown to be $T^{1/2}$-consistent and
asymptotically normal. Its asymptotic variance and standard error can be
constructed in the same manner as those of the joint GMM estimator $\hat{
\Greekmath 0112 }^{\ast}(\hat{A})$. In particular, a formula analogous to (\ref
{GIV_Est_STD}) applies after removing the components associated with the
supply-side moment conditions.
\end{remark}
\begin{remark}
The preceding discussion also extends naturally to specification testing. In
particular, the $J$-test constructed from the moment conditions in (\ref
{Moment_phi}) and the corresponding GIV estimator provides a test of the
validity of the demand-side moment restrictions. Under correct
specification, the resulting $J$-statistic converges in distribution to $
\Greekmath 011F ^{2}(n-\bar {r}-1)$. Analogous estimation, inference, and
specification-testing results hold when the analysis is based solely on the
supply-side moment conditions.
\end{remark}
\subsection{Estimating the number of GIVs}
Construction of the GIVs requires knowledge of the rank $\bar{r}$ of the
matrix $(\mathbf{1}_{n},\Greekmath 0115 )$, which may not be feasible in practice. In
this subsection, we propose a Bayesian information criterion (BIC) for
consistent estimation of $\bar{r}$. The construction is motivated by Lemma
\ref{ID_G_GIV_Weight} and Lemma \ref{G_GIV_Solutions}.
Specifically, let $\{ \hat{\Greekmath 0116 }_{j}\}_{j\leq n-1}$ denote the eigenvalues of
$\hat{S}_{y}$ arranged in increasing order. Define the information criterion
\begin{equation}
\mathrm{BIC}_{T}(j)\equiv \frac{T}{n-j}\sum_{s=1}^{n-j}\frac{(\hat{\Greekmath 0116 }_{s}-
\hat{\Greekmath 0116 }_{1})^{2}}{2\hat{\Greekmath 0116 }_{s}^{2}}+j\log(T), \label{BIC}
\end{equation}
for $j\in \mathcal{J}$, where $\mathcal{J}\equiv \{1,\ldots,n-1\}$. The
estimator $\hat{r}$ of $\bar{r}$ is then given by
\begin{equation}
\hat{r}=\arg \min_{j\in \mathcal{J}}\mathrm{BIC}_{T}(j). \label{r_hat}
\end{equation}
The consistency of $\hat{r}$ is established in Theorem \ref
{r_hat_consistency}.
\begin{theorem}
\label{r_hat_consistency}\ Under Assumptions \ref{ID}, \ref{Asy_Cond_1} and \ref{S}, we have $\hat{r}=\bar{r}$ with probability approaching 1.
\end{theorem}
We conclude this subsection by providing intuition for the construction of (
\ref{BIC}). The criterion $\mathrm{BIC}_{T}(j)$ consists of two components.
The first term, $T(n-j)^{-1}\sum_{s=1}^{n-j}(\hat{\Greekmath 0116 }_{s}-\hat{\Greekmath 0116 }
_{1})^{2}/(2\hat{\Greekmath 0116 }_{s}^{2})$, is decreasing in $j$ and captures an
over-fitting effect analogous to that in classical regression settings,
where $j$ reflects the dimension or complexity of the model. The second
term, $j\log(T)$, is strictly increasing in $j$ and serves as a penalty on
model complexity. The estimator $\hat{r}$ in (\ref{r_hat}) therefore
balances the trade-off between goodness-of-fit and model complexity.
The eigenvalues $\{ \hat{\Greekmath 0116 }_{j}\}_{j\leq n-1}$ are $T^{1/2}$-consistent
estimators of $\{ \Greekmath 0116 _{j}\}_{j\leq n-1}$ under Assumptions \ref{Asy_Cond_1}
(i, ii). Since $\Greekmath 0116 _{s}=\bar{\Greekmath 011B }_{u}^{2}$ for $s\in \{1,\ldots,n-\bar{r}
\}$, it follows that for $j\geq \bar{r}$,
\begin{equation*}
\frac{T}{n-j}\sum_{s=1}^{n-j}\frac{(\hat{\Greekmath 0116 }_{s}-\hat{\Greekmath 0116 }_{1})^{2}}{2\hat{
\Greekmath 0116 }_{s}^{2}}=O_{p}(1).
\end{equation*}
Consequently, the penalty term $j\log(T)$ dominates the first term in (\ref
{BIC}). Since $j\log(T)$ is strictly increasing in $j$, $\mathrm{BIC}_{T}(j)$
is asymptotically minimized at $\bar{r}$ over $j\in \{ \bar{r},\ldots,n-1\}$
. On the other hand, for $j<\bar{r}$, we have $n-j\geq n-\bar {r}+1$, so
Assumption \ref{Asy_Cond_1}(iii) implies that $\Greekmath 0116 _{n-j}-\Greekmath 0116 _{1}$ is bounded
away from zero. Combined with the $T^{-1/2}$ consistency of $\{ \hat{\Greekmath 0116 }
_{j}\}_{j\leq n-1}$, this yields
\begin{equation*}
\frac{T}{n-j}\sum_{s=1}^{n-j}\frac{(\hat{\Greekmath 0116 }_{s}-\hat{\Greekmath 0116 }_{1})^{2}}{2\hat{
\Greekmath 0116 }_{s}^{2}}\geq \frac{T}{n-j}\frac{(\hat{\Greekmath 0116 }_{n-j}-\hat{\Greekmath 0116 }_{1})^{2}}{2
\hat{\Greekmath 0116 }_{n-j}^{2}}\geq K^{-1}T((\Greekmath 0116 _{n-j}-\Greekmath 0116 _{1})^{2}-O_{p}(T^{-1/2})),
\end{equation*}
which diverges at rate $T$. This term therefore dominates the penalty term
of order $\log(T)$, implying that $\mathrm{BIC}_{T}(j)>\mathrm{BIC}_{T}(\bar{
r})$ wpa1 for all $j<\bar{r}$. Combining the two cases, $\mathrm{BIC}_{T}(j)$
is asymptotically minimized at $j=\bar{r}$ over $j\in \mathcal{J}$, which
ensures the consistency of $\hat{r}$.
\section{Extensions and Discussion\label{sec: Extension}}
This section provides further discussion and extensions of the main results
established in the previous section. First, we show that the identification
strategy in \cite{gabaix2024granular} may fail when the factor loadings are
unknown, potentially leading to inconsistent estimation and invalid
inference under standard GMM procedures. Second, we demonstrate that our
estimation and inference procedures can be straightforwardly extended to
settings with exogenous regressors in the demand and supply equations and to
data with unbalanced features. The latter extension shows that our methods
remain applicable even when the number of entities $n$ is large and may
exceed $T$.
\subsection{Identification failure of factor loadings in \protect\cite
{gabaix2024granular}\ \label{subsec: GK}}
In this subsection, we show that the identification strategy in \cite
{gabaix2024granular}, in particular their Proposition 7, may fail when $
\Greekmath 0115 $ is unknown. \cite{gabaix2024granular} impose a normalization on the
factor loadings $\Greekmath 0115 $ by setting the loadings of the first factor $
\Greekmath 0111 _{1,t}$ to be $\mathbf{1}_{n}$, and the loadings of the remaining
factors $\Greekmath 0111 _{2,t}$, denoted by $\Greekmath 0115 _{-1}$, to satisfy\footnote{
See the second paragraph above Proposition 7 in \cite{gabaix2024granular}.
In the same paragraph, they also impose the restriction that $\mathrm{Var}
(\Greekmath 0115 _{-1}^{\top}y_{t})$ is diagonal with distinct diagonal entries\ ($
\Greekmath 0115 _{-1}^{\top}y_{t}$ here is equal to $n\check{\Greekmath 0111 }_{t}$ in their
notation). As we show below, this additional restriction should be
interpreted as an assumption on the latent factors rather than a
normalization on the factor loadings. Moreover, imposing this restriction
does not resolve the identification issue in their approach.}
\begin{equation}
\mathbf{1}_{n}^{\top}\Greekmath 0115 _{-1}=\mathbf{0}_{r-1}^{\top}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, \ \ }
n^{-1}\Greekmath 0115 _{-1}^{\top}\Greekmath 0115 _{-1}=\mathbf{I}_{r-1}. \label{GK_F_0a}
\end{equation}
Moreover, they assume\footnote{
See the second paragraph on page 2279, display (3), and Assumption 3 in \cite
{gabaix2024granular}.}
\begin{equation}
\mathrm{Var}(u_{t})=\Greekmath 011B _{u}^{2}\mathbf{I}_{n},\qquad \mathrm{Cov}(\Greekmath 0111
_{t},u_{t})=\mathbf{0}_{r\times n},\qquad \mathbb{E}[\Greekmath 0111 _{t}]=\mathbf{0}
_{r},\qquad \mathrm{Cov}(u_{t},\Greekmath 0122 _{t})=\mathbf{0}_{n}.
\label{GK_F_0b}
\end{equation}
For notational simplicity, we abstract from exogenous regressors in both the
demand and supply equations. The demand equation (\ref{G_demand}) can then
be written as
\begin{equation}
y_{t}=\Greekmath 011E p_{t}\mathbf{1}_{n}+\Greekmath 0111 _{1,t}\mathbf{1}_{n}+\Greekmath 0115 _{-1}\Greekmath 0111
_{2,t}+u_{t}, \label{GK_F_1}
\end{equation}
while the supply equation remains the same as (\ref{G_supply}).
\cite{gabaix2024granular} propose using both $\Greekmath 0115 _{\bot}^{\top}y_{t}$
and $\Greekmath 0115 _{-1}^{\top}y_{t}$ as IVs to construct moment conditions for
identifying and estimating $\Greekmath 011E $, $\Greekmath 0120 $, and $\Greekmath 0115 _{-1}$.\footnote{
They correspond to $z_t(m^y)$ and $\check{\Greekmath 0111 }_t(m^y)$ in Proposition~4 of
\citet{gabaix2024granular}.
} Because $\mathbf{1}
_{n}^{\top}\Greekmath 0115 _{-1}=\mathbf{0}_{r-1}^{\top}$, one can partial out $
\Greekmath 0111 _{2,t}$ by premultiplying (\ref{GK_F_1}) by $e^{\top}$, yielding
\begin{equation}
y_{e,t}-\Greekmath 011E p_{t}=\Greekmath 0111 _{1,t}+u_{e,t}. \label{GK_F_2}
\end{equation}
Moreover, because $n^{-1}\Greekmath 0115 _{-1}^{\top}\Greekmath 0115 _{-1}=\mathbf{I}_{r-1}$,
premultiplying (\ref{GK_F_1}) by $\Greekmath 0115 _{-1}^{\top}$, we also obtain
\begin{equation}
\Greekmath 0115 _{-1}^{\top}y_{t}=n\Greekmath 0111 _{2,t}+\Greekmath 0115 _{-1}^{\top}u_{t}.
\label{GK_F_3}
\end{equation}
From Assumption \ref{ID}(i) and (\ref{GK_F_0b}), it follows that the product
of (\ref{GK_F_3}) and the second term on the right of (\ref{GK_F_2}) is such
that
\begin{equation}
\mathbb{E}[u_{e,t}\Greekmath 0115 _{-1}^{\top}y_{t}]=n\mathbb{E}[\Greekmath 0111 _{2,t}u_{e,t}]+
\Greekmath 0115 _{-1}^{\top}\mathbb{E}[u_{t}u_{e,t}]=\mathbf{0}_{r-1}.
\label{GK_F_4}
\end{equation}
However, the product $\mathbb{E}[\Greekmath 0111 _{1,t}\Greekmath 0115 _{-1}^{\top}y_{t}]$ of the
first term on the right of (\ref{GK_F_2}) and (\ref{GK_F_3}) may be nonzero
due to possible correlation between $\Greekmath 0111 _{1,t}$ and $\Greekmath 0111 _{2,t}$.
Therefore, $\Greekmath 0115 _{-1}^{\top}y_{t}$ cannot be directly used together with $
y_{e,t}-\Greekmath 011E p_{t}$ to form valid moment conditions. To address this issue,
\cite{gabaix2024granular} introduce the regression coefficient $b_{y}$ of $
\Greekmath 0111 _{1,t}$ on $y_{t}^{\top}\Greekmath 0115 _{-1}$ such that
\begin{equation}
\mathbb{E}[(\Greekmath 0111 _{1,t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{y})\Greekmath 0115 _{-1}^{
\top}y_{t}]=\mathbf{0}_{r-1}. \label{GK_F_4b}
\end{equation}
Combining this with (\ref{GK_F_2}) and (\ref{GK_F_4}) yields
\begin{equation}
\mathbb{E}[(y_{e,t}-\Greekmath 011E p_{t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{y})\Greekmath 0115
_{-1}^{\top}y_{t}]=\mathbf{0}_{r-1}. \label{GK_F_5a}
\end{equation}
Since $y_{e,t}-\Greekmath 011E p_{t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{y}=u_{e,t}-u_{t}^{\top
}\Greekmath 0115 _{-1}b_{y}+\Greekmath 0111 _{1,t}-n\Greekmath 0111 _{2,t}^{\top}b_{y}$, it follows from
Assumption \ref{ID}(i) and (\ref{GK_F_0b}) that
\begin{equation*}
\mathbb{E}\! \left[ (y_{e,t}-\Greekmath 011E
p_{t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{y})\Greekmath 0115 _{\bot}^{\top}y_{t}\right] =
\mathbb{E}\! \left[ (u_{e,t}-u_{t}^{\top}\Greekmath 0115 _{-1}b_{y}+\Greekmath 0111 _{1,t}-n
\Greekmath 0111 _{2,t}^{\top}b_{y})\Greekmath 0115 _{\bot }^{\top}u_{t}\right] =\mathbf{0}_{n-r}.
\end{equation*}
Together with (\ref{GK_F_5a}), this yields the moment conditions from the
demand equation:
\begin{equation}
\mathbb{E}\left[ (y_{e,t}-\Greekmath 011E p_{t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{y})\left(
\begin{array}{c}
\Greekmath 0115 _{\bot}^{\top}y_{t} \\
\Greekmath 0115 _{-1}^{\top}y_{t}
\end{array}
\right) \right] =\mathbf{0}_{n-1}. \label{GK_F_5}
\end{equation}
Similarly, $\Greekmath 0115 _{-1}^{\top}y_{t}$ cannot be directly used together with $
p_{t}-\Greekmath 0120 y_{S,t}$ to identify $\Greekmath 0120 $, because $\Greekmath 0115 _{-1}^{\top}y_{t}$
contains $\Greekmath 0111 _{2,t}$, which may be correlated with $\Greekmath 0122 _{t}$. \cite
{gabaix2024granular} therefore propose
\begin{equation*}
\mathbb{E}\! \left[ (p_{t}-\Greekmath 0120
y_{S,t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{p})\Greekmath 0115 _{-1}^{\top}y_{t}\right] =
\mathbf{0}_{r-1},
\end{equation*}
where $b_{p}$ is defined by
\begin{equation*}
\mathbb{E}\! \left[ (\Greekmath 0122 _{t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{p})
\Greekmath 0115 _{-1}^{\top}y_{t}\right] =\mathbf{0}_{r-1}.
\end{equation*}
Moreover, by Assumption \ref{ID}(i) and (\ref{GK_F_0b}),
\begin{equation*}
\mathbb{E}\! \left[ (p_{t}-\Greekmath 0120
y_{S,t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{p})\Greekmath 0115 _{\bot}^{\top}y_{t}\right] =
\mathbb{E}\! \left[ (\Greekmath 0122
_{t}-n\Greekmath 0111 _{2,t}^{\top}b_{p}-u_{t}^{\top}\Greekmath 0115 _{-1}b_{p})\Greekmath 0115 _{\bot
}^{\top}u_{t}\right] =\mathbf{0}_{n-r},
\end{equation*}
which provides additional moment conditions. Therefore, the moment
conditions from the supply equation are
\begin{equation}
\mathbb{E}\left[ (p_{t}-\Greekmath 0120 y_{S,t}-y_{t}^{\top}\Greekmath 0115 _{-1}b_{p})\left(
\begin{array}{c}
\Greekmath 0115 _{\bot}^{\top}y_{t} \\
\Greekmath 0115 _{-1}^{\top}y_{t}
\end{array}
\right) \right] =\mathbf{0}_{n-1}. \label{GK_F_6}
\end{equation}
The moment conditions in (\ref{GK_F_5}) and (\ref{GK_F_6}) coincide with
those in Proposition 4 of \cite{gabaix2024granular} when additional
exogenous variables are excluded. \cite{gabaix2024granular} argue in their
Proposition 4 that these conditions identify $\Greekmath 011E $, $\Greekmath 0120 $, $b_{y}$, and $
b_{p}$ when $\Greekmath 0115 _{-1}$ is known. When $\Greekmath 0115 _{-1}$ is unknown, they
propose the additional moment conditions
\begin{equation}
\mathbb{E}\big[(M_{\mathbf{1}_{n}}-\Greekmath 0115 _{-1}(\Greekmath 0115 _{-1}^{\top}
\Greekmath 0115 _{-1})^{-1}\Greekmath 0115 _{-1}^{\top})y_{t}y_{t}^{\top}\Greekmath 0115 _{-1}\big]=
\mathbf{0}_{n\times(r-1)}, \label{GK_F_7}
\end{equation}
which correspond to equation (50) in Proposition 7 of \cite
{gabaix2024granular}. To verify (\ref{GK_F_7}), note that $M_{\mathbf{1}
_{n}}\Greekmath 0115 _{-1}=\Greekmath 0115 _{-1}$ by (\ref{GK_F_0a}). Using (\ref{GK_F_1}),
\begin{equation*}
(M_{\mathbf{1}_{n}}-\Greekmath 0115 _{-1}(\Greekmath 0115 _{-1}^{\top}\Greekmath 0115 _{-1})^{-1}
\Greekmath 0115 _{-1}^{\top})y_{t}=(M_{\mathbf{1}_{n}}-\Greekmath 0115 _{-1}(\Greekmath 0115 _{-1}^{
\top }\Greekmath 0115 _{-1})^{-1}\Greekmath 0115 _{-1}^{\top})u_{t}.
\end{equation*}
Hence, (\ref{GK_F_7}) follows from Assumption \ref{ID}(i).
Since (\ref{GK_F_7}) provides $n(r-1)$ moment conditions for $n(r-1)$
unknown entries in $\Greekmath 0115 _{-1}$, it may seem to deliver exact
identification.\footnote{
Indeed, \cite{gabaix2024granular} state at the top of page 2294 that
\textquotedblleft The new moment (50) identifies $\check {\Greekmath 0115 }$." In our
notation, their moment (50) corresponds to (\ref{GK_F_7}), while their $
\check{\Greekmath 0115 }$ is denoted here by $\Greekmath 0115 _{-1}$.} However, as shown in
the lemma below, this is not the case: the restrictions in (\ref{GK_F_7})
fail to uniquely identify $\Greekmath 0115 _{-1}$, even up to rotation.
\begin{lemma}
\label{GK_Non_ID} Suppose that $\mathbb{E}[y_{t}y_{t}^{\top}]$ is finite and
nonsingular, and that the conditions in (\ref{GK_F_0b}) hold. Let $
\{d_{j}\}_{j=1}^{n-1}$ be an orthonormal basis of eigenvectors associated
with the nonzero eigenvalues of $\mathbb{E}[\tilde{y}_{t}\tilde{y}_{t}^{\top
}]$. For any subset $J\subset \{1,\ldots,n-1\}$ with $|J|=r-1$, let $D_{J}$
collect the columns $d_{j}$, $j\in J$. Then $n^{1/2}D_{J}$ satisfies (\ref
{GK_F_0a}) and (\ref{GK_F_7}).
\end{lemma}
Lemma \ref{GK_Non_ID} shows that the moment condition (\ref{GK_F_7}),
together with the normalization in (\ref{GK_F_0a}), fails to identify $\func{
col}(\Greekmath 0115 _{-1})$. Indeed, (\ref{GK_F_0a}) and (\ref{GK_F_7}) admit $
\binom{n-1}{r-1}$ different choices of $D_{J}$, whose column spaces are
generally distinct. Moreover,
\begin{equation*}
\func{col}((d_{1},\ldots,d_{n-1}))=\func{col}(M_{\mathbf{1}_{n}})=\func{col}
((\Greekmath 0115 _{-1},\Greekmath 0115 _{\bot})).
\end{equation*}
Therefore, if $\func{col}(D_{J})\neq \func{col}(\Greekmath 0115 _{-1})$, then the
orthogonal complement of $(\mathbf{1}_{n},D_{J})$, denoted by $D_{J,\bot}$,
need not be orthogonal to the true factor space $\func{col}(\Greekmath 0115 _{-1})$.
In particular,
\begin{equation*}
D_{J,\bot}^{\top}\Greekmath 0115 _{-1}\neq \mathbf{0}_{(n-r)\times(r-1)}
\end{equation*}
may hold. Consequently, the candidate GIVs $D_{J,\bot}^{\top}y_{t}$ may
still contain components of the latent factors $\Greekmath 0111 _{t}$, since
\begin{equation*}
D_{J,\bot}^{\top}y_{t}=D_{J,\bot}^{\top}\Greekmath 0115 _{-1}\Greekmath 0111 _{2,t}+D_{J,\bot
}^{\top}u_{t}.
\end{equation*}
As a result, moment conditions constructed from $D_{J,\bot}^{\top}y_{t}$ may
fail to eliminate the latent factor component and therefore need not provide
valid identifying restrictions for the structural parameters.
To make the identification problem more explicit, consider the special case $
r=n-1$. In this case, $\Greekmath 0115 _{\bot}$ is one-dimensional, and the moment
conditions in (\ref{GK_F_5}) and (\ref{GK_F_6}) provide only exact
identification for the unknown parameters $\Greekmath 011E $, $\Greekmath 0120 $, $b_{y}$, and $b_{p}
$, given the true factor loading matrix $\Greekmath 0115 _{-1}$ and its orthogonal
complement $\Greekmath 0115 _{\bot}$. Therefore, identification of these parameters
ultimately relies on identification of $\func{col}(\Greekmath 0115 _{-1})$.
However, as discussed above, (\ref{GK_F_5}) and (\ref{GK_F_6}) admit $n-1$
different choices of $D_{J}$, and hence $n-1$ corresponding choices of $
D_{J,\bot}$. Since $\Greekmath 0115 _{\bot}$ is one-dimensional, at least $n-2$ of
these choices satisfy
\begin{equation*}
D_{J,\bot}^{\top}\Greekmath 0115 _{-1}\neq \mathbf{0}_{1\times(r-1)}.
\end{equation*}
For such choices, the corresponding candidate GIVs $D_{J,\bot}^{\top}y_{t}$
retain latent factor components and therefore generally fail to identify the
true elasticities $\Greekmath 011E $ and $\Greekmath 0120 $. Consequently, GMM estimation based on
these invalid instruments would generally converge to pseudo-true values
rather than the true structural parameters. Moreover, since different
choices of $D_{J,\bot}$ generally lead to different pseudo-true values, the
resulting limits need not even be uniquely determined.
Lemma \ref{GK_Non_ID} establishes the non-identification of the factor
loadings under the normalization in (\ref{GK_F_0a}). In addition to (\ref
{GK_F_0a}), \cite{gabaix2024granular} also assume that $\mathrm{Var}
(\Greekmath 0115 _{-1}^{\top}y_{t})$ is a diagonal matrix with distinct diagonal
entries.\footnote{
See the second paragraph above Proposition 7 in \cite{gabaix2024granular}.}
Under (\ref{GK_F_0a}) and (\ref{GK_F_0b}), however,
\begin{equation*}
\mathrm{Var}(\Greekmath 0115 _{-1}^{\top}y_{t})=\Greekmath 0115 _{-1}^{\top}\mathbb{E}
[y_{t}y_{t}^{\top}]\Greekmath 0115 _{-1}=n^{2}\mathbb{E}[\Greekmath 0111 _{2,t}\Greekmath 0111 _{2,t}^{\top
}]+n\Greekmath 011B _{u}^{2}\mathbf{I}_{r-1}.
\end{equation*}
Hence, the diagonal structure imposed on $\mathrm{Var}(\Greekmath 0115 _{-1}^{\top
}y_{t})$ amounts to additional restrictions on the latent factors, requiring
that the components of $\Greekmath 0111 _{2,t}$ are uncorrelated and have distinct
variances. Since $\mathbb{E}[y_{t}y_{t}^{\top}]$ is unknown, this condition
should be viewed as an assumption on the latent factors rather than a
restriction on the factor loadings.\footnote{
If one instead imposes the corresponding restriction on the sample second
moment $T^{-1}\sum_{t\leq T}y_{t}y_{t}^{\top}$, then for a broad class of
data-generating processes, the orthonormalized eigenvectors associated with
any collection of $r-1$ eigenvalues may satisfy the empirical counterpart of
this restriction in finite samples due to estimation error in $
T^{-1}\sum_{t\leq T}y_{t}y_{t}^{\top}$.}
Even after imposing this restriction together with (\ref{GK_F_0a}), the
moment condition (\ref{GK_F_7}) still fails to identify $\func{col}
(\Greekmath 0115 _{-1})$. Indeed, by (\ref{GK_F_0a}), (\ref{GK_F_0b}), and (\ref
{GK_F_1}),
\begin{equation*}
\mathbb{E}[\tilde{y}_{t}\tilde{y}_{t}^{\top}]=M_{\mathbf{1}_{n}}\mathbb{E}
[y_{t}y_{t}^{\top}]M_{\mathbf{1}_{n}}=\Greekmath 0115 _{-1}\mathbb{E}
[\Greekmath 0111 _{2,t}\Greekmath 0111 _{2,t}^{\top}]\Greekmath 0115 _{-1}^{\top}+\Greekmath 011B _{u}^{2}M_{\mathbf{1}
_{n}}.
\end{equation*}
Under the additional assumption that $\mathbb{E}[\Greekmath 0111 _{2,t}\Greekmath 0111 _{2,t}^{\top}]
$ is diagonal with distinct diagonal entries, the matrix $\mathbb{E}[\tilde {
y}_{t}\tilde{y}_{t}^{\top}]$ has $r-1$ distinct eigenvalues larger than $
\Greekmath 011B _{u}^{2}$, while $\Greekmath 011B _{u}^{2}$ itself is an eigenvalue with
multiplicity $n-r$.\ The eigenspace associated with $\Greekmath 011B _{u}^{2}$ is
spanned by $\Greekmath 0115 _{\bot}$. Consequently, there are at least $(r-1)(n-r)+1$ such
choices of $D_{J}$ with distinct column spaces.\footnote{
Note that one choice is given by $D_{J}=n^{-1/2}\Greekmath 0115 _{-1}$, while the
remaining $(r-1)(n-r)$ choices are obtained by selecting $r-2$ columns from $
\Greekmath 0115 _{-1}$ columns and one column from $\Greekmath 0115 _{\bot}$.} Since these
columns are orthonormal eigenvectors of $\mathbb{E}[\tilde{y}_{t}\tilde{y}
_{t}^{\top}]$, it follows that
\begin{equation*}
\mathrm{Var}(n^{1/2}D_{J}^{\top}y_{t})=nD_{J}^{\top}\mathbb{E}
[y_{t}y_{t}^{\top}]D_{J}=nD_{J}^{\top}\mathbb{E}[\tilde{y}_{t}\tilde{y}
_{t}^{\top }]D_{J}
\end{equation*}
is diagonal with distinct diagonal entries. Since there exist $(r-1)(n-r)+1$
\ such choices of $D_{J}$ with distinct column spaces, this again shows that
$\func{col}(\Greekmath 0115 _{-1})$ is not identified.
We conclude this subsection with a lemma establishing the rotational
non-uniqueness of the factor loadings identified from (\ref{GK_F_0a}), (\ref
{GK_F_5})--(\ref{GK_F_7}) in the general case.
\begin{lemma}
\label{GK_Rotation} Suppose that $\{b_{y},b_{p},\Greekmath 0115 _{-1},\Greekmath 0115 _{\perp
}\}$ satisfies (\ref{GK_F_5})--(\ref{GK_F_7}) given $\Greekmath 011E $ and $\Greekmath 0120 $. Let $
C_{1}$ and $C_{2}$ be arbitrary orthogonal matrices of dimensions $
(r-1)\times(r-1)$ and $(n-r)\times(n-r)$, respectively. Define
\begin{equation*}
\Greekmath 0115 _{C_{1}}\equiv \Greekmath 0115 _{-1}C_{1},\qquad \Greekmath 0115 _{C_{2}}\equiv
\Greekmath 0115 _{\perp}C_{2}.
\end{equation*}
Then $\{C_{1}^{\top}b_{y},C_{1}^{\top}b_{p},\Greekmath 0115 _{C_{1}},\Greekmath 0115 _{C_{2}}\}
$ also satisfies (\ref{GK_F_5})--(\ref{GK_F_7}) given $\Greekmath 011E $ and $\Greekmath 0120 $.
Moreover, $\Greekmath 0115 _{C_{1}}$ and $\Greekmath 0115 _{C_{2}}$ satisfy the same
normalization and orthogonality restrictions as $\Greekmath 0115 _{-1}$ and $
\Greekmath 0115 _{\perp}$.
\end{lemma}
Lemma \ref{GK_Rotation} shows that the factor loadings $\Greekmath 0115 _{-1}$ and
the regression coefficients $b_{y}$ and $b_{p}$ are, at best, identified up
to an orthonormal rotation. This raises concerns for estimation and
inference based on these moment conditions, since standard GMM procedures
require uniqueness of the identified parameters and are therefore not
directly applicable in this setting.
The non-identification issue here is more challenging to address than in our
approach, because $\{b_{y},b_{p},\Greekmath 0115 _{-1},\Greekmath 0115 _{\perp}\}$ are jointly
identified together with the demand and supply elasticities. This joint
determination complicates both the computation of the GIV estimator and the
analysis of its statistical properties. In contrast, our approach separates
the identification of the factor loadings from that of the elasticity
parameters. We then exploit the invariance properties of the GIV estimator
and the $J$-test statistic to address the fact that the factor loadings are
only identified up to rotation, thereby allowing standard GMM estimation and
inference to remain valid.
\subsection{Model with exogenous regressors\label{subsec: Ex1}}
This subsection extends the model studied in the previous section by
allowing for additional exogenous regressors in both the demand and supply
equations in (\ref{G_demand})--(\ref{G_supply}). Specifically, we consider
\begin{align}
y_{t} & =\Greekmath 011E p_{t}\mathbf{1}_{n}+x_{t}\Greekmath 010C +\Greekmath 0115 \Greekmath 0111 _{t}+u_{t},
\label{F_demand} \\
p_{t} & =\Greekmath 0120 y_{S,t}+w_{t}^{\top}\Greekmath 010D +\Greekmath 0122 _{t}, \label{F_supply}
\end{align}
where $x_{t}\equiv(x_{1,t},\ldots,x_{n,t})^{\top}$ with $x_{i,t}\in \mathbb{R
}^{d_{x}}$, and $w_{t}\in \mathbb{R}^{d_{w}}$ denote observed exogenous
variables that have direct effects on demand and supply, respectively. The
variables $x_{t}$ include both sector fixed effects and unit-level demand
shifters, while $w_{t}$ captures aggregate supply shifters.\footnote{
Since $\Greekmath 0115 \Greekmath 0111 _{t}$ can be decomposed as $\Greekmath 0115 (\Greekmath 0111 _{t}-\mathbb{E}
[\Greekmath 0111 _{t}])+\Greekmath 0115 \mathbb{E}[\Greekmath 0111 _{t}]$, and $\Greekmath 0115 \mathbb{E}[\Greekmath 0111 _{t}]$
can be absorbed into the sector fixed effects, we assume without loss of
generality that $\mathbb{E}[\Greekmath 0111 _{t}]=\mathbf{0}_{r}$ throughout this
subsection.} Under suitable exogeneity conditions, the main identification
and estimation arguments continue to apply after partialling out $x_{t}$
from the demand equation (\ref{F_demand}).\footnote{
When $x_{t}$ includes variables excluded from the supply equation, these may
serve as IVs for identifying $\Greekmath 0120 $ in (\ref{F_supply}) if they are
uncorrelated with $\Greekmath 0122 _{t}$. Similarly, variables in $w_{t}$
excluded from the demand equation may identify $\Greekmath 011E $ in (\ref{F_demand}) if
they are uncorrelated with $\Greekmath 0111 _{t}$ and $u_{t}$. Although standard in the
classical simultaneous equations literature, this strategy is not widely
used in empirical applications of GIV. We therefore do not assume the
existence or exogeneity of such excluded variables.}
Specifically, multiplying $M_{\mathbf{1}_{n}}$ on both sides of (\ref
{F_demand}) yields
\begin{equation*}
\tilde{y}_{t}=\tilde{x}_{t}\Greekmath 010C +\tilde{\Greekmath 0115 }\Greekmath 0111 _{t}+\tilde{u}_{t},
\end{equation*}
where $\tilde{x}_{t}\equiv M_{\mathbf{1}_{n}}x_{t}$, and $\tilde{y}_{t}$, $
\tilde{\Greekmath 0115 }$, and $\tilde{u}_{t}$ are defined analogously; see (\ref
{tilda_vars}). Let $\tilde{y}_{t}^{\ast}\equiv M_{\mathbf{1}_{n}}y_{t}^{\ast}
$, where $y_{t}^{\ast}\equiv y_{t}-x_{t}\Greekmath 010C $. Then the above equation can
be written as
\begin{equation*}
\tilde{y}_{t}^{\ast}=\tilde{\Greekmath 0115 }\Greekmath 0111 _{t}+\tilde{u}_{t},
\end{equation*}
which takes a form similar to (\ref{Demeaned_Demand}).\footnote{
Since $w_{t}$ is invariant across $i$, it is automatically partialled out in
$\tilde{y}$. Therefore, $\tilde{y}_{t}^{\ast}$ effectively partials out the
exogenous regressors in both the demand and supply equations.} Therefore, we
can apply Lemmas \ref{ID_G_GIV_Weight} and \ref{G_GIV_Solutions} in
Subsection \ref{subsec: G_model_ID}, with the second moment matrix $\bar{
\Sigma}_{\tilde{y}}$ replaced by
\begin{equation*}
\bar{\Sigma}_{\tilde{y}^{\ast}}\equiv T^{-1}\sum_{t\leq T}\mathbb{E}[\tilde {
y}_{t}^{\ast}\tilde{y}_{t}^{\ast \top}],
\end{equation*}
to identify the subspace $\func{col}(\bar{\Greekmath 0115 }_{\bot})$, which is
orthogonal to $(\mathbf{1}_{n},\Greekmath 0115 )$.
Given $\Greekmath 010C $, the GIVs and moment conditions can be constructed in the same
way as in the previous section, with $y_{t}$ replaced by $y_{t}^{\ast}$. To
proceed, we need to estimate $\Greekmath 010C $, which is required to construct an
estimator for $y_{t}^{\ast}$. The unknown parameter $\Greekmath 010C $ is estimated by
\begin{equation*}
\hat{\Greekmath 010C }\equiv \Bigl(\sum_{t\leq T}\tilde{x}_{t}^{\top}\tilde{x}_{t}\Bigr)
^{-1}\Bigl(\sum_{t\leq T}\tilde{x}_{t}^{\top}\tilde{y}_{t}\Bigr).
\end{equation*}
If $x_{t}$ is exogenous, in the sense that it is uncorrelated with both $
\Greekmath 0111 _{t}$ and $u_{t}$, then standard least squares theory implies that $\hat{
\Greekmath 010C }$ is a $T^{1/2}$-consistent estimator of $\Greekmath 010C $. Given $\hat{\Greekmath 010C }$
, define
\begin{equation*}
\hat{y}_{t}^{\ast}\equiv y_{t}-x_{t}\hat{\Greekmath 010C },\qquad \hat{y}_{e,t}^{\ast
}\equiv e^{\top}\hat{y}_{t}^{\ast}.
\end{equation*}
We then obtain $\hat{A}\equiv Q_{-1}\hat{A}_{0}$, where $\hat{A}_{0}$
collects the eigenvectors corresponding to the smallest\ $n-\bar{r}$
eigenvalues of
\begin{equation*}
\hat{S}_{y^{\ast}}\equiv Q_{-1}^{\top}\hat{\Sigma}_{\hat{y}
^{\ast}}Q_{-1},\qquad \hat{\Sigma}_{\hat{y}^{\ast}}\equiv T^{-1}\sum_{t\leq
T}\hat {y}_{t}^{\ast}\hat{y}_{t}^{\ast \top}.
\end{equation*}
The moment conditions used to estimate the unknown parameter $\Greekmath 0112
\equiv(\Greekmath 011E ,\Greekmath 0120 ,\Greekmath 010D ^{\top})^{\top}$ are
\begin{equation}
\bar{g}_{T}(\Greekmath 0112 ;\hat{A},\hat{\Greekmath 010C })\equiv T^{-1}\sum_{t\leq T}
\begin{pmatrix}
\hat{A}^{\top}\hat{y}_{t}^{\ast}(\hat{y}_{e,t}^{\ast}-\Greekmath 011E p_{t}) \\
\hat{A}^{\top}\hat{y}_{t}^{\ast}(p_{t}-\Greekmath 0120 y_{S,t}-w_{t}^{\top}\Greekmath 010D ) \\
w_{t}(p_{t}-\Greekmath 0120 y_{S,t}-w_{t}^{\top}\Greekmath 010D )
\end{pmatrix}
. \label{F_Moments}
\end{equation}
The GIV estimator is defined as
\begin{equation}
\hat{\Greekmath 0112 }(\hat{A})\equiv \arg \min_{\Greekmath 0112 \in \Theta}\bar{g}_{T}(\Greekmath 0112 ;
\hat{A},\hat{\Greekmath 010C })^{\top}W_{0,T}(\hat{A})\bar{g}_{T}(\Greekmath 0112 ;\hat{A},\hat{
\Greekmath 010C }), \label{G_GIV_1}
\end{equation}
where
\begin{equation}
W_{0,T}(\hat{A})\equiv \Bigl(\mathrm{diag}(\hat{A}^{\top},\hat{A}^{\top },
\mathbf{I}_{d_{w}})\,W_{0,T}\, \mathrm{diag}(\hat{A},\hat{A},\mathbf{I}
_{d_{w}})\Bigr)^{-1}, \label{F_Weight}
\end{equation}
and $W_{0,T}$ is a user-specified symmetric positive definite $
(2n+d_{w})\times(2n+d_{w})$ matrix. Since $\bar{g}_{T}(\Greekmath 0112 ;\hat{A},\hat{
\Greekmath 010C })$ is linear in $\Greekmath 0112 $, the GIV estimator takes the same form as in (
\ref{GIV_Form_1}), with $D_{j,T}(\hat{A})$ redefined as
\begin{equation}
D_{j,T}(\hat{A})\equiv \mathrm{diag}(\hat{A}^{\top},\hat{A}^{\top},\mathbf{I}
_{d_{w}})D_{j,T},\quad j=1,2, \label{GIV_Form_3}
\end{equation}
where
\begin{equation}
D_{1,T}\equiv T^{-1}\sum_{t\leq T}
\begin{pmatrix}
\hat{y}_{t}^{\ast}p_{t} & \mathbf{0}_{n} & \mathbf{0}_{n\times d_{w}} \\
\mathbf{0}_{n} & \hat{y}_{t}^{\ast}y_{S,t} & \hat{y}_{t}^{\ast}w_{t}^{\top}
\\
\mathbf{0}_{d_{w}} & w_{t}y_{S,t} & w_{t}w_{t}^{\top}
\end{pmatrix}
,\quad D_{2,T}\equiv T^{-1}\sum_{t\leq T}
\begin{pmatrix}
\hat{y}_{t}^{\ast}\hat{y}_{e,t}^{\ast} \\
\hat{y}_{t}^{\ast}p_{t} \\
w_{t}p_{t}
\end{pmatrix}
. \label{GIV_Form_4}
\end{equation}
To conduct inference on $\Greekmath 0112 $ and test the validity of the moment
conditions in (\ref{F_Moments}), one must account for the estimation error
in $\hat{\Greekmath 010C }$, since it enters $\bar{g}_{T}(\Greekmath 0112 ;\hat{A},\hat{\Greekmath 010C })$
through both $\hat{y}_{t}^{\ast}$ and $\hat{A}$. Lemma \ref{Moment_est} in
Online Appendix \ref{APP_5} shows that the randomness introduced by the
estimation error of $\hat{\Greekmath 010C }$ is of higher order. Therefore, the
estimation error of $\hat{\Greekmath 010C }$ is asymptotically negligible and can be
ignored. Consequently, the standard errors of $\hat{\Greekmath 0112 }(\hat{A})$ and
the specification tests can be constructed in the same way as in the
previous section. See Algorithm 1 in Online Appendix \ref{APP_0} for details.
\subsection{Model with unbalanced data structure\label{subsec: Ex2}}
The data used to estimate the demand and supply equations have, thus far,
been assumed to follow a balanced structure.\ For instance, $y_{i,t}$
denotes the demand of entity $i$ in period $t$, where $i\in \{1,\ldots,n\}$
and $t\in \{1,\ldots,T\}$. The analysis in the previous section assumes a
balanced data structure, so that each time period is associated with the
same number of entities. In practice, however, entry and exit lead to an
unbalanced data structure. As we show below, the identification and
estimation approach extends naturally to this setting, provided that the
entry and exit decisions of entities are independent of their demand. \
Specifically, the demand equation in (\ref{G_demand}) is generalized as
\begin{equation*}
y_{t}=\Greekmath 011E p_{t}\mathbf{1}_{n_{t}}+\Greekmath 0115 \Greekmath 0111 _{t}+u_{t},
\end{equation*}
where $y_{t}\equiv(y_{i,t})_{i\leq n_{t}}$ and $n_{t}$ denotes the number of
entities present in the market at time $t$. The factor-loading matrix $
\Greekmath 0115 $ is of dimension $n_t\times r$, and the idiosyncratic shock vector
$u_t$ is of dimension $n_t\times1$. The supply equation in (\ref{G_supply})
remains unchanged, with $y_{S,t}\equiv S_{t}^{\top}y_{t}$, where $S_{t}$ is
an $n_{t}\times1$ vector of market shares.
To construct the GIV, consider a subsample of $n_{0}$ entities that are
observed in all periods. Let $y_{t}^{0}$ denote the corresponding subvector
of $y_{t}$. Their demand equation is
\begin{equation}
y_{t}^{0}=\Greekmath 011E p_{t}\mathbf{1}_{n_{0}}+\Greekmath 0115 _{0}\Greekmath 0111 _{t}+u_{t}^{0},
\label{n0_Demand}
\end{equation}
where $\Greekmath 0115 _{0}$ and $u_{t}^{0}$ are the associated submatrices of $
\Greekmath 0115 $ and $u_{t}$. Let $Q_{0,-1}\equiv(q_{0,2},\ldots,q_{0,n_{0}})\in
\mathbb{R}^{n_{0}\times(n_{0}-1)}$ be defined analogously to (\ref{ortho_q_j}
) with $n$ replaced by $n_{0}$, and $\Greekmath 0115 _{0,\bot}\in \mathbb{R}
^{n_{0}\times(n_{0}-\bar{r}_{0})}$ denotes the orthogonal complement of $(
\mathbf{1}_{n_{0}},\Greekmath 0115 _{0})$ where $\bar{r}_{0}\equiv \mathrm{rank}((
\mathbf{1}_{n_{0}},\Greekmath 0115 _{0}))$. Since $\Greekmath 0115
_{0,\bot}^{\top}y_{t}^{0}=\Greekmath 0115 _{0,\bot}^{\top}u_{t}^{0}$, Assumption \ref
{ID} implies
\begin{equation*}
\mathbb{E}[\Greekmath 0111 _{t}u_{t}^{0\top}\Greekmath 0115 _{0,\bot}]=\mathbf{0}_{r\times (n_{0}-
\bar{r}_{0})},\quad \mathbf{1}_{n_{t}}^{\top}\mathbb{E}[u_{t}u_{t}^{0\top}]
\Greekmath 0115 _{0,\bot}=\mathbf{0}_{n_{0}-\bar{r}_{0}},\quad \mathbb{E}
[\Greekmath 0122 _{t}u_{t}^{0\top}\Greekmath 0115 _{0,\bot}]=\mathbf{0}_{n_{0}-\bar{r}
_{0}}.
\end{equation*}
These imply $\Greekmath 0115 _{0,\bot}^{\top}y_{t}^{0}$ provides valid moment
conditions
\begin{equation*}
\mathbb{E}\! \left[
\begin{pmatrix}
\Greekmath 0115 _{0,\bot}^{\top}y_{t}^{0}(y_{e,t}-\Greekmath 011E p_{t}) \\
\Greekmath 0115 _{0,\bot}^{\top}y_{t}^{0}(p_{t}-\Greekmath 0120 y_{S,t})
\end{pmatrix}
\right] =\mathbf{0}_{2(n_{0}-\bar{r}_{0})},
\end{equation*}
which identify $\Greekmath 011E $ and $\Greekmath 0120 $.
Lemmas \ref{ID_G_GIV_Weight} and \ref{G_GIV_Solutions} apply to this
subsample. In particular, $\Greekmath 0115 _{0,\bot}$ can be consistently estimated
(up to an orthonormal rotation) by $\hat{A}\equiv Q_{0,-1}\hat{A}_{0}$,
where $\hat{A}_{0}$ collects the eigenvectors corresponding to the smallest $
n_{0}-\bar{r}_{0}$ eigenvalues of
\begin{equation*}
\hat{S}_{y^{0}}\equiv Q_{0,-1}^{\top}\hat{\Sigma}_{y^{0}}Q_{0,-1},\qquad
\hat{\Sigma}_{y^{0}}\equiv T^{-1}\sum_{t\leq T}y_{t}^{0}y_{t}^{0\top}.
\end{equation*}
The GIV estimator is defined analogously to (\ref{GMM_Criterion}), with
moment function
\begin{equation*}
\bar{g}_{T}^{0}(\Greekmath 0112 ;\hat{A})\equiv T^{-1}\sum_{t\leq T}
\begin{pmatrix}
\hat{A}^{\top}y_{t}^{0}(y_{e,t}-\Greekmath 011E p_{t}) \\
\hat{A}^{\top}y_{t}^{0}(p_{t}-\Greekmath 0120 y_{S,t})
\end{pmatrix}
,
\end{equation*}
and a user-specified symmetric positive definite $2n_{0}\times2n_{0}$ weight
matrix, and it admits an explicit form
\begin{equation}
\hat{\Greekmath 0112 }(\hat{A})=\big(D_{1,T}^{0}(\hat{A})^{\top}W_{0,T}(\hat{A}
)D_{1,T}^{0}(\hat{A})\big)^{-1}\big(D_{1,T}^{0}(\hat{A})^{\top}W_{0,T}(\hat {
A})D_{2,T}^{0}(\hat{A})\big), \label{GIV_Form_n0_1}
\end{equation}
where $D_{j,T}^{0}(\hat{A})\equiv(\mathbf{I}_{2}\otimes \hat{A}
^{\top})D_{j,T}^{0}$ for $j=1,2$, and
\begin{equation}
D_{1,T}^{0}\equiv T^{-1}\sum_{t\leq T}\mathrm{diag}\! \left(
y_{t}^{0}p_{t},\,y_{t}^{0}y_{S,t}\right) ,\qquad D_{2,T}^{0}\equiv
T^{-1}\sum_{t\leq T}
\begin{pmatrix}
y_{t}^{0}y_{e,t} \\
y_{t}^{0}p_{t}
\end{pmatrix}
. \label{GIV_Form_n0_2}
\end{equation}
The asymptotic normality of the GIV estimator and the asymptotic
distribution of the $J$-test statistic follow from the same arguments as in
the proofs of Theorems \ref{Asy_Dist} and \ref{J_Test}, with the appropriate
modifications to the $D_{1}$ and $V$ matrices. To conserve space, the
implementation details of the GIV estimation and inference procedure are
provided in Algorithm 2 in Online Appendix \ref{APP_0}.
\begin{remark}
The method developed in this subsection constructs GIVs using data from a
subset of $n_{0}$ entities. Once the GIVs are obtained, the full data set
can be used to construct moment conditions for estimating the unknown
parameters in the model. Since $n_{0}$ may be substantially smaller than
both $n$ and $T$, the proposed approach can be applied in settings where the
cross-sectional dimension is large and may even exceed the sample size.
\end{remark}
\begin{remark}
The flexibility of using only a subset of entities to construct the GIVs
also allows the framework to accommodate heterogeneous demand elasticities
across entities, provided that a subset of entities is known to share a
common elasticity.
Specifically, suppose that the first $n_{0}$ entities share a common demand
elasticity $\bar{\Greekmath 011E }$. We may use the demand equations for these entities,
i.e., (\ref{n0_Demand}), to construct the GIVs $\hat{A}^{\top}y_{t}^{0}$,
which can then be employed to form the moment functions
\begin{equation}
\bar{g}_{T}^{n_{0}}(\bar{\Greekmath 011E };\hat{A})\equiv T^{-1}\sum_{t\leq T}\hat {A}
^{\top}y_{t}^{0}(y_{n_{0},t}-\bar{\Greekmath 011E }p_{t})\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ \ \ and \ \ }\bar {g}
_{T}^{i}(\Greekmath 011E _{i};\hat{A})\equiv T^{-1}\sum_{t\leq T}\hat{A}
^{\top}y_{t}^{0}(y_{i,t}-\Greekmath 011E _{i}p_{t}), \label{GIV_Form_n0_3}
\end{equation}
to estimate the common elasticity $\bar{\Greekmath 011E }$ and the entity-specific
elasticities $\Greekmath 011E _{i}$ for $i=n_{0}+1,\ldots,n$, where $
y_{n_{0},t}=n_{0}^{-1}\mathbf{1}_{n_{0}}^{\top}y_{t}^{0}$. Using arguments
analogous to those in the proof of Theorem \ref{Asy_Dist}, it can be shown
that the resulting GIV estimators of $\bar{\Greekmath 011E }$ and $\Greekmath 011E _{i}$ ($
i=n_{0}+1,\ldots,n$) are $T^{1/2}$-consistent and asymptotically normal.
Together with consistent estimators of their asymptotic variances, these
results can be used to conduct inference on heterogeneous demand
elasticities and to test hypotheses such as $H_{0}:\bar{\Greekmath 011E }=\Greekmath 011E _{i}$ and $
H_{0}:\Greekmath 011E _{i}=\Greekmath 011E _{i^{\prime}}$ for $i\neq i^{\prime}$.
\end{remark}
\section{Simulation Studies\label{sec: MC}}
We examine the finite-sample performance of the proposed GIV estimation and
inference procedures through Monte Carlo experiments. Subsection \ref
{sec-mc1} describes the simulation design, and Subsection \ref{sec-mc2}
reports the results.
\subsection{Simulation Setting\ \label{sec-mc1}}
We consider two simulation designs, a baseline design and an extended
design, to investigate the finite-sample performance of the proposed GIV
estimator, the associated inference procedures, and the specification test.
The baseline design follows the model in (\ref{G_demand})--(\ref{G_supply}),
while the extended design augments the demand equation with three exogenous
regressors.
Solving the demand-supply system in (\ref{F_demand})--(\ref{F_supply}), with
exogenous regressors in the demand equation but no additional exogenous
variables in the supply equation, yields the following reduced-form
expressions under the extended design:
\begin{align}
y_{t} & =\left( \mathbf{I}_{n}+\frac{\Greekmath 011E \Greekmath 0120 }{1-\Greekmath 011E \Greekmath 0120 }\mathbf{1}
_{n}S_{t}^{\top}\right) (x_{t}\Greekmath 010C +\Greekmath 0115 \Greekmath 0111 _{t}+u_{t})+\frac{\Greekmath 011E }{
1-\Greekmath 011E \Greekmath 0120 }\mathbf{1}_{n}\Greekmath 0122 _{t}, \label{F_Reduced_Demand} \\
p_{t} & =\frac{\Greekmath 0120 }{1-\Greekmath 011E \Greekmath 0120 }S_{t}^{\top}(x_{t}\Greekmath 010C +\Greekmath 0115 \Greekmath 0111
_{t}+u_{t})+\frac{\Greekmath 0122 _{t}}{1-\Greekmath 011E \Greekmath 0120 }. \label{F_Reduced_Supply}
\end{align}
To generate the simulated data, we first draw the demand and supply shocks $
\Greekmath 0111 _{t}$, $u_{t}$, and $\Greekmath 0122 _{t}$, the exogenous regressors $x_{t}$,
and the market share vector $S_{t}$ conditional on the parameter values of $
\Greekmath 011E $, $\Greekmath 0120 $, $\Greekmath 010C $, and $\Greekmath 0115 $. These simulated values are then
substituted into (\ref{F_Reduced_Demand})--(\ref{F_Reduced_Supply}) to
obtain $(p_{t},y_{t})$. This procedure generates the simulated observations $
\{y_{t},p_{t},x_{t},S_{t}\}$ for each period $t$.
The demand and supply shocks are mutually independent and i.i.d.\ across $t$
, with
\begin{equation}
\Greekmath 0111 _{t}\sim N(\mathbf{0}_{r+1},\Sigma_{\Greekmath 0111 }),\qquad u_{t}\sim N(\mathbf{0}
_{n},\mathbf{I}_{n}),\qquad \Greekmath 0122 _{t}\sim
N(0,\Greekmath 011B _{\Greekmath 0122 }^{2}), \label{MC_Shocks}
\end{equation}
where $\Sigma_{\Greekmath 0111 }\equiv((0.1)^{|i-j|})_{i,j\leq r+1}$, and $\Greekmath 011B
_{\Greekmath 0122 }^{2}$ is set to $0.5$. The demand and supply elasticities are
set to $\Greekmath 011E =-0.5$ and $\Greekmath 0120 =1.5$, respectively. The exogenous regressors $
\{x_{i,t}\}$ are generated i.i.d.\ from $N(\mathbf{0}_{3},\mathbf{I}_{3})$
across $i$ and $t$, independently of $(\Greekmath 0111 _{t}^{\top},u_{t}^{\top
},\Greekmath 0122 _{t})^{\top}$. The market share vector $S_{t}$ is fixed across $
t$ and follows a Pareto rank-size specification:\ \ $s_{i}\propto \left(
i/n\right) ^{-1/\Greekmath 0116 _{S}}$, with tail index $\Greekmath 0116 _{S}=0.2$, where the shares
are normalized to satisfy $\sum_{i=1}^{n}s_{i}=1$. Such a power-law profile
is consistent with the size distributions documented for industries, firms,
and financial intermediaries \citep{gabaix2011granular,gabaix2024granular},
and generates the concentrated cross-sectional structure under which
granular variation is informative.
The factor loading matrix is specified as $\Greekmath 0115 =(\mathbf{1}
_{n},\Greekmath 0115 _{-1})$, where $\Greekmath 0115 _{-1}$ is the $n\times r$ matrix of
loadings on the $r$ non-aggregate latent factors. To construct $\Greekmath 0115 _{-1}$
, we first draw $nr$ independent $N(0,1)$ random variables to form an $
n\times r$ matrix $\Greekmath 0115 _{0}$. We then project $\Greekmath 0115 _{0}$ onto the
orthogonal complement of $(\mathbf{1}_{n},S)$ and obtain a preliminary
loading matrix $\tilde{\Greekmath 0115 }_{0}$ through the QR decomposition $M_{(
\mathbf{1}_{n},S)}\Greekmath 0115 _{0}=\tilde{\Greekmath 0115 }_{0}R_{\Greekmath 0115 _{0}}$,\ where $
R_{\Greekmath 0115 _{0}}$ is an $r\times r$ upper triangular matrix and the columns
of $\tilde{\Greekmath 0115 }_{0}$ are orthonormal. We then set $\Greekmath 0115 _{-1}=n^{1/2}
\tilde{\Greekmath 0115 }_{0}$. By construction, the resulting loading matrix $
\Greekmath 0115 =(\mathbf{1}_{n},\Greekmath 0115 _{-1})$ has rank $\bar{r}=r+1$.
We set $\Greekmath 010C =\mathbf{0}_{d_{x}}$ in (\ref{F_Reduced_Demand})--(\ref
{F_Reduced_Supply}) for both designs. In the baseline design, $\Greekmath 010C $ is
treated as known and the regressors $x_{t}$ are omitted. In the extended
design, however, $\Greekmath 010C $ is estimated using the procedure described in
Subsection~\ref{subsec: Ex1}. Because $\Greekmath 010C =\mathbf{0}_{d_{x}}$, the
regressors $x_{t}$ play no role in the data-generating process. Thus, any
difference between the two designs reflects the additional estimation error
associated with estimating $\Greekmath 010C $ and partialling out $x_{t}$.
We consider six combinations of the number of entities and the number of
non-aggregate latent factors:
\begin{equation}
(n,r)\in \{(5,1),(5,2),(8,3),(8,5),(10,5),(10,7)\}, \label{MC_nr_pair}
\end{equation}
together with three sample sizes, $T\in \{150,300,450\}$.\ To evaluate the
finite-sample performance of the GIV estimator, as well as the size
properties of the associated inference and specification tests, we conduct $
10{,}000$ Monte Carlo replications for each $(n,r,T)$ cell under each
simulation design.
To assess the effect of estimation error arising from the recovery of the
subspace orthogonal to $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$ on the
performance of the GIV estimator, we consider an oracle GIV estimator
constructed under the assumption that $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$
is known. Consequently, the matrix $A$, whose columns form a basis for the
orthogonal complement of $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$ and are used
to construct the GIVs, is treated as known.\footnote{
In the simulation, $A$ is constructed from the left singular vectors
associated with the zero singular values in the singular value decomposition
of $(\mathbf{1}_{n},\Greekmath 0115 )$.}\ The oracle GIV estimator therefore bypasses
both the BIC step for estimating the number of factors and the estimation of
the orthogonal complement of $\func{col}((\mathbf{1}_{n},\Greekmath 0115 ))$. In the
extended design, the oracle estimator additionally treats $\Greekmath 010C $ as known
and partials out $x_{t}\Greekmath 010C $ using the true parameter value. By contrast,
the feasible GIV estimator selects $\bar {r}$ using the BIC criterion in (
\ref{BIC})--(\ref{r_hat}), and then constructs the GIVs and the
corresponding GIV estimator according to Algorithm 1 of Online Appendix \ref
{APP_0}.
Both the oracle and feasible GIV estimators are evaluated using their
finite-sample root mean squared errors (RMSEs), with the results reported in
Table \ref{tab:rmse} of the next subsection. We also investigate the
empirical rejection probabilities of the two-sided tests of $H_{0}:\Greekmath 011E =-0.5$
and $H_{0}:\Greekmath 0120 =1.5$ at the $5\%$ significance level. The results are
reported in Table \ref{tab:t_size} of the next subsection. Inference is
conducted using $t$-tests based on the standard error estimators described
in Algorithm 1 of Online Appendix \ref{APP_0}, together with the asymptotic
normality of the GIV estimators.
To evaluate the power of the $J$-test, we consider a controlled violation of
the covariance restrictions underlying GIV validity while keeping the
factor-loading matrix $\Greekmath 0115 =(\mathbf{1}_{n},\Greekmath 0115 _{-1})$, the
structural parameters, and the marginal distributions of $u_{t}$, $\Greekmath 0111 _{t}$
, and $\Greekmath 0122 _{t}$ unchanged. Let $b_{n}=Ad_{n}$, where $d_{n}\in
\mathbb{R}^{n-r-1}$ is the unit vector obtained by applying the
Gram--Schmidt procedure to the first standard basis vector against $A^{\top}S
$. For $\Greekmath 011A \in \lbrack0,1)$, we generate $u_{t}$ as
\begin{equation*}
u_{t}=u_{t}^{\ast}+\frac{\Greekmath 011A }{\Greekmath 011B _{\Greekmath 0122 }}\Greekmath 0122 _{t}b_{n}-
\Bigl(1-(1-\Greekmath 011A ^{2})^{1/2}\Bigr)\bigl(b_{n}^{\top}u_{t}^{\ast }\bigr)b_{n},
\end{equation*}
where $u_{t}^{\ast}\sim N(\mathbf{0}_{n},\mathbf{I}_{n})$ is generated
together with $\Greekmath 0122 _{t}$ and $\Greekmath 0111 _{t}$ in the same way as $u_{t}$ in
(\ref{MC_Shocks}).
By construction, $b_{n}\in \func{col}(A)$ implies $b_{n}^{\top }\mathbf{1}
_{n}=0\ $and$\ b_{n}^{\top}\Greekmath 0115 _{-1}=\mathbf{0}_{r}^{\top}$, so $b_{n}$
is orthogonal to $\func{col}(\Greekmath 0115 )$. It is straightforward to verify that
for any $\Greekmath 011A \in \lbrack0,1)$, the joint distribution of $
(u_{t},\Greekmath 0122 _{t})$ remains Gaussian with
\begin{equation*}
\func{Var}(u_{t})=\mathbf{I}_{n}\qquad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and}\qquad \func{Cov}
(u_{t},\Greekmath 0122 _{t})=\Greekmath 011A \Greekmath 011B _{\Greekmath 0122 }b_{n}.
\end{equation*}
When $\Greekmath 011A =0$, the design reduces to the correctly specified benchmark
design used above to study the finite-sample properties of the GIV
estimators and inference procedures. When $\Greekmath 011A >0$, however, $u_{t}$ and $
\Greekmath 0122 _{t}$ become correlated, with heterogeneous correlation patterns
determined by $b_{n}$. As a result, the orthogonality conditions underlying
the GIVs are violated, rendering the GIVs invalid. Moreover, the degree of
misspecification increases linearly with $\Greekmath 011A $.
We use the six $(n,r)$ combinations in (\ref{MC_nr_pair}) together with the
three sample sizes $T$ considered above, vary $\Greekmath 011A $ over the grid $\Greekmath 011A
_{j}=0.02j\ $for $j=0,\ldots,20$, and conduct $10{,}000$ simulation
replications for each $(n,r,T,\Greekmath 011A )$ cell under each design. The
significance level of the $J$-test is set at $0.05$, and the resulting
empirical rejection probabilities are reported in Figure~\ref
{fig:j_power_baseline} of the next subsection.
\begin{table}[!t]
\caption{Root Mean Squared Error of the GIV Estimators}
\label{tab:rmse}\centering
{\tmpsmall\sc \setlength{\tabcolsep}{4.5pt} \begin{threeparttable}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cc cccc cccc@{}}
\toprule
& & \multicolumn{4}{c}{Baseline ($d_x = 0$)}
& \multicolumn{4}{c}{Extended ($d_x = 3$)} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
& & \multicolumn{2}{c}{Oracle} & \multicolumn{2}{c}{Feasible}
& \multicolumn{2}{c}{Oracle} & \multicolumn{2}{c}{Feasible} \\
\cmidrule(lr){3-4}\cmidrule(lr){5-6}\cmidrule(lr){7-8}\cmidrule(lr){9-10}
$(n, r)$ & $T$
& $\Greekmath 011E $ & $\Greekmath 0120 $ & $\Greekmath 011E $ & $\Greekmath 0120 $
& $\Greekmath 011E $ & $\Greekmath 0120 $ & $\Greekmath 011E $ & $\Greekmath 0120 $ \\
\midrule
\multirow{3}{*}{$(5, 1)$}
& 150 & 0.129 & 0.086 & 0.130 & 0.087 & 0.126 & 0.085 & 0.131 & 0.091 \\
& 300 & 0.088 & 0.060 & 0.092 & 0.065 & 0.088 & 0.060 & 0.089 & 0.065 \\
& 450 & 0.072 & 0.049 & 0.076 & 0.051 & 0.071 & 0.049 & 0.074 & 0.050 \\
\addlinespace
\multirow{3}{*}{$(5, 2)$}
& 150 & 0.128 & 0.086 & 0.150 & 0.109 & 0.129 & 0.086 & 0.160 & 0.116 \\
& 300 & 0.089 & 0.060 & 0.106 & 0.078 & 0.088 & 0.061 & 0.103 & 0.083 \\
& 450 & 0.071 & 0.049 & 0.094 & 0.072 & 0.071 & 0.049 & 0.095 & 0.071 \\
\addlinespace
\multirow{3}{*}{$(8, 3)$}
& 150 & 0.118 & 0.082 & 0.119 & 0.084 & 0.118 & 0.082 & 0.121 & 0.084 \\
& 300 & 0.081 & 0.057 & 0.081 & 0.058 & 0.080 & 0.058 & 0.081 & 0.059 \\
& 450 & 0.066 & 0.047 & 0.067 & 0.048 & 0.066 & 0.046 & 0.067 & 0.052 \\
\addlinespace
\multirow{3}{*}{$(8, 5)$}
& 150 & 0.118 & 0.082 & 0.142 & 0.115 & 0.117 & 0.082 & 0.152 & 0.115 \\
& 300 & 0.080 & 0.057 & 0.100 & 0.070 & 0.081 & 0.057 & 0.111 & 0.081 \\
& 450 & 0.066 & 0.046 & 0.081 & 0.062 & 0.066 & 0.046 & 0.097 & 0.080 \\
\addlinespace
\multirow{3}{*}{$(10, 5)$}
& 150 & 0.113 & 0.082 & 0.115 & 0.084 & 0.115 & 0.081 & 0.118 & 0.083 \\
& 300 & 0.079 & 0.057 & 0.080 & 0.057 & 0.080 & 0.057 & 0.081 & 0.058 \\
& 450 & 0.064 & 0.046 & 0.064 & 0.046 & 0.064 & 0.045 & 0.064 & 0.046 \\
\addlinespace
\multirow{3}{*}{$(10, 7)$}
& 150 & 0.113 & 0.080 & 0.145 & 0.113 & 0.113 & 0.081 & 0.142 & 0.110 \\
& 300 & 0.079 & 0.056 & 0.102 & 0.083 & 0.080 & 0.056 & 0.107 & 0.093 \\
& 450 & 0.064 & 0.045 & 0.081 & 0.058 & 0.064 & 0.045 & 0.081 & 0.059 \\
\bottomrule
\end{tabular*}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes.} This table reports the root mean squared errors of the GIV estimators based on
$10{,}000$ Monte Carlo replications, separately for the oracle and feasible
estimators, under the baseline design ($d_x=0$) and the extended design with
$d_x=3$ exogenous regressors. The oracle estimator treats $\Greekmath 0115 $ (and
$\Greekmath 010C $ when present) as known. The feasible estimator selects $\bar r$ using
the BIC criterion in~\eqref{BIC}--\eqref{r_hat} and is computed according to
Algorithm~1 in Online Appendix~\ref{APP_0}. Both estimators are evaluated using the
same simulated data within each replication. To reduce the influence of extreme draws, the RMSE is computed using the truncated loss function $\min \{5,(\widehat{\Greekmath 010B }-\Greekmath 010B )^2\}$ for a generic estimator $\widehat{\Greekmath 010B }$ of the parameter $\Greekmath 010B $.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\subsection{Simulation Results \label{sec-mc2}}
Table \ref{tab:rmse} shows that the RMSEs of both the oracle and feasible
estimators of $\Greekmath 011E $ and $\Greekmath 0120 $ decline substantially as the sample size $T$
increases. For example, in the baseline design with $(n,r)=(5,1)$, the RMSE
of the oracle estimator of $\Greekmath 011E $ decreases from $0.129$ at $T=150$, to $
0.088$ at $T=300$, and further to $0.072$ at $T=450$. Similar improvements
are observed across all configurations and for both structural parameters.
The feasible GIV estimator closely tracks the oracle estimator throughout
the simulation designs. In most cases, the difference in RMSE between the
oracle and feasible estimators is negligible. For instance, in the baseline
design with $(n,r)=(10,5)$, the RMSEs of the feasible estimator $\hat{\Greekmath 011E }
^{\ast }(\hat{A})$ are $0.115$, $0.080$, and $0.064$ at $T=150$, $300$, and $
450$, respectively, compared with the corresponding oracle RMSEs of $0.113$,
$0.079$, and $0.064$. Similar qualitative patterns are observed for the GIV
estimator of $\Greekmath 0120 $, as well as for both estimators in the extended design
with exogenous regressors.
We next examine the size properties of the $t$-tests for $H_{0}:\Greekmath 011E =-0.5$
and $H_{0}:\Greekmath 0120 =1.5$ at the $5\%$ significance level. The results, reported
in Table \ref{tab:t_size}, show that the empirical rejection probabilities
approach the nominal level as the sample size $T$ increases. When the sample
size is relatively small, i.e., $T=150$, the tests exhibit modest
over-rejection, with the distortion becoming more pronounced in
configurations involving a larger number of GIVs. For example, in the
baseline design with $(n,r)=(8,3)$, where there are four GIVs and eight
moment conditions, the empirical rejection probabilities of the $t$-tests
based on the feasible GIV estimators for $\Greekmath 011E $ and $\Greekmath 0120 $ are $0.104$ and $
0.073$, respectively, at $T=150$. By contrast, when $(n,r)=(8,5)$, where
only two GIVs are available, the corresponding rejection probabilities are $
0.075$ and $0.061$, respectively. Similar patterns are observed in the
extended design.
The over-rejection observed in these $t$-tests does not appear to be
primarily driven by estimation error in the number of factors or in the null
space of $(\mathbf{1}_{n},\Greekmath 0115 )$, since the tests based on the oracle GIV
estimators, which do not require estimation of these nuisance parameters,
display similar finite-sample behavior. Instead, the over-rejection is
likely related to the well-known many-moment bias in two-step GMM
estimation, of which the GIV estimator is a special case; see, for example,
\citet{HansenHeatonYaron1996} and \citet{newey2009generalized}. Several
approaches may help mitigate the resulting size distortion in small samples.
For example, instead of estimating $\Greekmath 011E $ and $\Greekmath 0120 $ jointly, one may
estimate them using two separate GMM procedures. This reduces the number of
moment conditions used in each estimation problem, although it may sacrifice
some of the efficiency gains from joint GMM estimation. Another possibility
is to employ the continuously updated GMM estimator rather than the two-step
GMM estimator. While this approach may improve finite-sample inference, it
also introduces additional computational burden, since the continuously
updated GMM estimator does not admit a closed-form solution.
\begin{table}[!t]
\caption{Empirical Rejection Probabilities of the Two-sided $t$-tests}
\label{tab:t_size}\centering
{\tmpsmall\sc \setlength{\tabcolsep}{4.5pt} \begin{threeparttable}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cc cccc cccc@{}}
\toprule
& & \multicolumn{4}{c}{Baseline ($d_x = 0$)}
& \multicolumn{4}{c}{Extended ($d_x = 3$)} \\
\cmidrule(lr){3-6}\cmidrule(lr){7-10}
& & \multicolumn{2}{c}{Oracle} & \multicolumn{2}{c}{Feasible}
& \multicolumn{2}{c}{Oracle} & \multicolumn{2}{c}{Feasible} \\
\cmidrule(lr){3-4}\cmidrule(lr){5-6}\cmidrule(lr){7-8}\cmidrule(lr){9-10}
$(n, r)$ & $T$
& $\Greekmath 011E $ & $\Greekmath 0120 $ & $\Greekmath 011E $ & $\Greekmath 0120 $
& $\Greekmath 011E $ & $\Greekmath 0120 $ & $\Greekmath 011E $ & $\Greekmath 0120 $ \\
\midrule
\multirow{3}{*}{$(5, 1)$}
& 150 & 0.091 & 0.063 & 0.094 & 0.062 & 0.089 & 0.058 & 0.090 & 0.061 \\
& 300 & 0.068 & 0.058 & 0.067 & 0.059 & 0.071 & 0.058 & 0.071 & 0.059 \\
& 450 & 0.063 & 0.052 & 0.065 & 0.053 & 0.058 & 0.056 & 0.059 & 0.058 \\
\addlinespace
\multirow{3}{*}{$(5, 2)$}
& 150 & 0.071 & 0.058 & 0.072 & 0.058 & 0.073 & 0.057 & 0.076 & 0.059 \\
& 300 & 0.062 & 0.052 & 0.064 & 0.053 & 0.060 & 0.057 & 0.062 & 0.056 \\
& 450 & 0.056 & 0.052 & 0.056 & 0.052 & 0.058 & 0.056 & 0.057 & 0.055 \\
\addlinespace
\multirow{3}{*}{$(8, 3)$}
& 150 & 0.101 & 0.069 & 0.104 & 0.073 & 0.104 & 0.070 & 0.113 & 0.073 \\
& 300 & 0.076 & 0.059 & 0.078 & 0.060 & 0.073 & 0.062 & 0.076 & 0.064 \\
& 450 & 0.068 & 0.058 & 0.068 & 0.059 & 0.067 & 0.055 & 0.071 & 0.056 \\
\addlinespace
\multirow{3}{*}{$(8, 5)$}
& 150 & 0.070 & 0.058 & 0.075 & 0.061 & 0.069 & 0.059 & 0.075 & 0.061 \\
& 300 & 0.058 & 0.058 & 0.058 & 0.056 & 0.057 & 0.054 & 0.059 & 0.053 \\
& 450 & 0.056 & 0.052 & 0.059 & 0.054 & 0.055 & 0.051 & 0.057 & 0.051 \\
\addlinespace
\multirow{3}{*}{$(10, 5)$}
& 150 & 0.096 & 0.072 & 0.102 & 0.077 & 0.101 & 0.067 & 0.107 & 0.073 \\
& 300 & 0.078 & 0.059 & 0.079 & 0.059 & 0.077 & 0.061 & 0.082 & 0.063 \\
& 450 & 0.068 & 0.058 & 0.068 & 0.057 & 0.068 & 0.053 & 0.066 & 0.054 \\
\addlinespace
\multirow{3}{*}{$(10, 7)$}
& 150 & 0.069 & 0.057 & 0.073 & 0.064 & 0.068 & 0.057 & 0.075 & 0.062 \\
& 300 & 0.060 & 0.057 & 0.061 & 0.059 & 0.061 & 0.051 & 0.062 & 0.053 \\
& 450 & 0.057 & 0.053 & 0.058 & 0.056 & 0.055 & 0.053 & 0.057 & 0.054 \\
\bottomrule
\end{tabular*}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes.} This table reports the empirical rejection rates of the two-sided $t$-tests of $H_{0}:\Greekmath 011E =-0.5$ and $H_{0}:\Greekmath 0120 =1.5$ at the 5\% nominal significance level, using the asymptotic critical value $1.96$, based on $10{,}000$ Monte Carlo replications. Results are reported separately for the oracle and feasible estimators under both the baseline design ($d_x=0$) and the extended design with $d_x=3$ exogenous regressors. The oracle estimator treats $\Greekmath 0115 $ (and $\Greekmath 010C $ when present) as known. The feasible estimator selects $\bar{r}$ using the BIC criterion in \eqref{BIC}--\eqref{r_hat} and is implemented according to Algorithm~1 in Online Appendix~\ref{APP_0}.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\begin{table}[!t]
\caption{Empirical Rejection Probabilities of the $J$-test}
\label{tab:j_size}\centering
{\tmpsmall\sc \setlength{\tabcolsep}{6pt} \begin{threeparttable}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cc cc cc@{}}
\toprule
& & \multicolumn{2}{c}{Baseline ($d_x = 0$)}
& \multicolumn{2}{c}{Extended ($d_x = 3$)} \\
\cmidrule(lr){3-4}\cmidrule(lr){5-6}
$(n, r)$ & $T$ & Oracle & Feasible & Oracle & Feasible \\
\midrule
\multirow{3}{*}{$(5, 1)$}
& 150 & 0.067 & 0.071 & 0.068 & 0.068 \\
& 300 & 0.060 & 0.062 & 0.058 & 0.059 \\
& 450 & 0.053 & 0.052 & 0.054 & 0.055 \\
\addlinespace
\multirow{3}{*}{$(5, 2)$}
& 150 & 0.060 & 0.061 & 0.059 & 0.061 \\
& 300 & 0.056 & 0.058 & 0.052 & 0.055 \\
& 450 & 0.056 & 0.057 & 0.058 & 0.057 \\
\addlinespace
\multirow{3}{*}{$(8, 3)$}
& 150 & 0.074 & 0.082 & 0.073 & 0.082 \\
& 300 & 0.064 & 0.064 & 0.062 & 0.067 \\
& 450 & 0.058 & 0.058 & 0.060 & 0.061 \\
\addlinespace
\multirow{3}{*}{$(8, 5)$}
& 150 & 0.060 & 0.064 & 0.055 & 0.063 \\
& 300 & 0.057 & 0.058 & 0.055 & 0.058 \\
& 450 & 0.050 & 0.050 & 0.053 & 0.057 \\
\addlinespace
\multirow{3}{*}{$(10, 5)$}
& 150 & 0.069 & 0.080 & 0.074 & 0.083 \\
& 300 & 0.055 & 0.060 & 0.064 & 0.069 \\
& 450 & 0.056 & 0.061 & 0.053 & 0.058 \\
\addlinespace
\multirow{3}{*}{$(10, 7)$}
& 150 & 0.060 & 0.065 & 0.055 & 0.063 \\
& 300 & 0.054 & 0.058 & 0.052 & 0.057 \\
& 450 & 0.054 & 0.056 & 0.053 & 0.054 \\
\bottomrule
\end{tabular*}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes.} This table reports the empirical rejection rates of the $J$-test at the $5\%$
nominal significance level based on $10{,}000$ Monte Carlo replications.
Results are reported separately for the oracle and feasible estimators under
both the baseline design ($d_x=0$) and the extended design with
$d_x=3$ exogenous regressors. For the oracle estimator, the reference
distribution is $\Greekmath 011F ^{2}_{2(n-r-2)}$. For the feasible estimator, the
reference distribution is the replication-specific
$\Greekmath 011F ^{2}_{2(n-\widehat{r}-1)}$. The oracle estimator treats $\Greekmath 0115 $ (and
$\Greekmath 010C $ when present) as known. The feasible estimator selects $\bar r$ using
the BIC criterion in~\eqref{BIC}--\eqref{r_hat} and is implemented according
to Algorithm~1 in Online Appendix~\ref{APP_0}.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\begin{figure}[!t]
\caption{Empirical Power of the $J$-test: Baseline Design}
\label{fig:j_power_baseline}\centering
\includegraphics[width=0.85
\textwidth]{./figures/power_figure_baseline_nominal.eps}
\begin{minipage}{0.95\textwidth}
\footnotesize
\noindent \textit{Notes.} This figure plots the empirical rejection probabilities of the $J$-test at the 5\% nominal significance level in the baseline design ($d_x=0$), based on $10{,}000$ Monte Carlo replications. Each panel corresponds to one of the six configurations $(n,r)$ listed in~\eqref{MC_nr_pair}, and within each panel three curves correspond to the sample sizes $T\in \{150,300,450\}$. The horizontal axis is $\Greekmath 011A $ and the vertical axis is the rejection rate. The horizontal dashed line marks the nominal $5\%$ level. Power is reported for the feasible GIV estimator, which selects $\bar{r}$ using the BIC criterion in~\eqref{BIC}--\eqref{r_hat} and is implemented according to Algorithm~1 in Online Appendix~\ref{APP_0}.
\end{minipage}
\end{figure}
Finally, we examine the performance of the $J$-test for assessing the
validity of the moment conditions constructed using the GIVs. Table~\ref
{tab:j_size} reports the empirical rejection probabilities at the $5\%$
nominal significance level under correct specification. The size behavior is
broadly similar to that of the $t$-tests reported in Table~\ref{tab:t_size}:
the empirical rejection probabilities approach the nominal level as the
sample size $T$ increases, while modest over-rejection is observed in small
samples, particularly when the number of moment conditions is relatively
large. For example, in the baseline design with $(n,r)=(10,5)$, the
empirical rejection probabilities of the oracle and feasible $J$-tests are $
0.069$ and $0.080$, respectively, at $T=150$. In the corresponding extended
design, the rejection probabilities are $0.074$ and $0.083$, respectively.
Similar patterns are observed for $(n,r)=(8,3)$, where the feasible
rejection probabilities are $0.082$ in both the baseline and extended
designs at $T=150$. As the sample size increases, the empirical rejection
probabilities move steadily toward the nominal $5\%$ level across all
configurations.
As discussed earlier for the $t$-tests, the observed small-sample
over-rejection does not appear to be primarily driven by estimation error in
the number of factors or in the null space of $(\mathbf{1}_{n},\Greekmath 0115 )$,
since the oracle and feasible procedures display very similar finite-sample
behavior. Instead, the distortion is likely related to the many-moment
nature of the GMM problem.
We next examine the power of the feasible $J$-test under misspecification.
Figure~\ref{fig:j_power_baseline} plots the empirical rejection
probabilities of the $J$-test as a function of $\Greekmath 011A $, which controls the
severity of the violation of the GIV moment conditions in the baseline
design.\footnote{
The corresponding results for the extended design are reported in Online
Appendix~\ref{APP_6} and exhibit similar qualitative patterns.} By
construction, each power curve begins near the nominal $5\%$ level when $
\Greekmath 011A =0$. The rejection probabilities then increase monotonically with $\Greekmath 011A $
, and the power curves become substantially steeper as the sample size $T$
increases. For example, in the baseline design with $(n,r)=(10,7)$, the test
achieves approximately $80\%$ power at $\Greekmath 011A \approx0.29$ when $T=150$, $
\Greekmath 011A \approx0.22$ when $T=300$, and $\Greekmath 011A \approx0.16$ when $T=450$. Similar
patterns are observed across all configurations. The feasible $J$
-test exhibits good power in detecting moderate violations of the moment
restrictions at empirically relevant sample sizes.
\section{Empirical Application: Aggregate Market Multiplier\label{sec:emp}}
A central question in asset pricing is how strongly the aggregate stock
market responds to shifts in investor demand for equities. This response is
summarized by the aggregate market multiplier, denoted by $
\Greekmath 0114 \equiv-\Greekmath 011E ^{-1}$, where $\Greekmath 011E $ is the aggregate demand elasticity.
Economically, $\Greekmath 0114 $ measures the change in aggregate equity value induced
by a one-dollar demand shock. The aggregate demand elasticity has become a
central object of interest in asset pricing and macro-finance because
investor demand, portfolio reallocation, and market segmentation can have
important effects on equilibrium asset prices; see, among others,
\citet{piazzesi2007asset}, \citet{koijen2019demand}, and
\citet{gabaix2021search}. In frictionless benchmark models, aggregate demand
is highly elastic, implying a multiplier close to zero. By contrast, the
inelastic-markets hypothesis predicts a substantially larger value of $\Greekmath 0114
$.
The size of the aggregate multiplier remains controversial. Standard
asset-pricing models imply a macro elasticity of roughly 10 to 20,
corresponding to a multiplier of only 0.05 to 0.1.\footnote{
See Appendices F and I of \citet{gabaix2021search} for computations of the
macro elasticity implied by the model of \citet{lucas1978asset}, the
rare-disaster models of \citet{barro2006rare} and \citet{gabaix2012variable}
, and the long-run risks model of \citet{bansal2004risks}.} Yet empirical
estimates of stock-level, factor-level, and aggregate demand elasticities
generally point to substantially less elastic demand. For example,
\citet{lou2012flow} estimates a stock-level multiplier of about 1.2,
\citet{pavlova2023benchmarking} report multipliers between 0.3 and 0.5, and \citet{gabaix2021search} report substantially larger multipliers at the aggregate level. These
findings are consistent with the economic intuition that aggregate equity
demand should be less elastic than demand for individual stocks, since
stocks are closer substitutes for one another than for alternative asset
classes such as bonds. At the same time, the estimated multipliers are an
order of magnitude larger than those implied by standard asset-pricing
models, posing a challenge for conventional theories of asset demand.
Resolving this discrepancy requires credible identification of the aggregate
demand elasticity, a key parameter for quantifying the effects of capital
flows, institutional demand shocks, and policy interventions on asset prices.
Obtaining such identification is challenging because prices and quantities
are jointly determined in equilibrium. Demand shocks affect prices, while
prices simultaneously enter investors' demand equations, rendering simple
regressions of demand on prices generally inconsistent. Building on the
demand-based asset-pricing framework of \citet{koijen2019demand},
\citet{gabaix2021search} address this endogeneity problem by exploiting the
latent-factor structure in investor demand and estimate aggregate
multipliers ranging from 4.73 to 5.85, with a median estimate close to five.
Their approach, however, relies on consistent estimation of the latent
demand factors and therefore on asymptotic arguments in which the
cross-sectional dimension diverges; see, for example,
\citet{bai2003inferential}. In this section, we revisit the aggregate
multiplier using the GIV framework developed in Section~\ref{sec: G_model},
which permits valid estimation and inference without requiring the number of
sectors to grow with the sample size.
Following \cite{gabaix2021search}, we study the aggregate multiplier through
a demand system for U.S. equity holdings. Investors are grouped into $n$
equity-holding sectors.\ Let $\Delta q_{i,t}$ denote the fractional
quarterly change in investor $i$'s equity holdings at quarter $t$, with its
empirical counterpart defined in\ \eqref{eq:dq} below; see Online Appendix
\ref{subsec:variable} for construction details. The demand equation is given
by
\begin{equation}
\Delta q_{i,t}=\Greekmath 011E \Delta p_{t}+\Greekmath 0115 _{i}^{\top}\Greekmath 0111 _{t}+u_{i,t} ,
\label{eq:gk-demand}
\end{equation}
where $\Delta p_{t}$ denotes the quarterly equity market return, $
\Greekmath 0111 _{t}\in \mathbb{R}^{r}$ represents latent aggregate demand factors, and $
u_{i,t}$ is an idiosyncratic demand shock. The parameter $\Greekmath 011E $ captures the
aggregate demand elasticity and is the primary object of interest.\footnote{
Unlike \citet{gabaix2021search}, we do not explicitly include aggregate macroeconomic variables, such as GDP growth, in (\ref{eq:gk-demand}). Aggregate variables that enter the demand equation with homogeneous loadings across sectors are eliminated by the orthogonality condition defining the GIVs, since the instruments are constructed to be orthogonal to $\mathbf{1}_n$. Aggregate variables with heterogeneous loadings are absorbed into the latent factor component $\Greekmath 0115 \Greekmath 0111 _t$. Under the maintained assumption that the idiosyncratic shocks $u_{i,t}$ are orthogonal to these aggregate components, their omission does not affect identification, estimation, or inference for the demand elasticity $\Greekmath 011E $ and the aggregate multiplier $\Greekmath 0114 $.
}
Because the aggregate supply of equity is approximately fixed in the short
run, market clearing implies that size-weighted net demand equals zero:
\begin{equation}
q_{S,t}\equiv S_{t}^{\top}\Delta q_{t}=0, \label{eq:gk-market_clearing}
\end{equation}
where $\Delta q_{t}=(\Delta q_{i,t})_{i\le n}$ and $S_{t}$ denotes the
vector of predetermined market shares. Combining \eqref{eq:gk-demand} and
\eqref{eq:gk-market_clearing} yields the equilibrium price equation
\begin{equation}
\Delta p_{t} = \Greekmath 0114 \left( S_{t}^{\top}\Greekmath 0115 \Greekmath 0111 _{t}+u_{S,t}\right), \label{eq:gk-equil}
\end{equation}
where $u_{S,t}\equiv S_{t}^{\top}u_{t}$. Thus, the aggregate multiplier $
\Greekmath 0114 $ measures the equilibrium price response to aggregate demand shocks,
with a less elastic demand (smaller $|\Greekmath 011E |$) corresponding to a larger multiplier.
Our data are drawn from the Financial Accounts of the United States,
Table~L.224, which reports the equity holdings of major investor sectors.
\footnote{
We use the June 2026 vintage of the Financial Accounts, in which corporate
equity holdings by sector are reported in Table~L.224.} Following \cite
{gabaix2021search}, our benchmark analysis uses the sample period
1993Q1--2018Q4 and includes twelve sectors that hold U.S. equities
continuously throughout this period. Table~\ref{tab:sectors} in Online
Appendix~\ref{app:sectors} lists these sectors together with their average
market shares. In the data, the fractional change in sector $i$'s equity
holdings is measured as
\begin{equation}
\Delta q_{i,t}\equiv \frac{w_{i,t}}{w_{i,t-1}R_{t}}-1, \label{eq:dq}
\end{equation}
where $w_{i,t}$ denotes the value of sector $i$'s equity holdings, $R_{t}
$ is the gross capital-appreciation return on the aggregate stock market,
and $\Delta p_{t}$ is measured by the quarterly simple return on the CRSP
value-weighted index excluding dividends.\footnote{
Online Appendix~\ref{subsec:variable} details the construction of $\Delta
q_{i,t}$ and $\Delta p_{t}$, and Online Appendix~\ref{subsec:sector}
describes the sector classification and market-share weights $S_{t}$.
Following \cite{gabaix2021search}, pooled sector-level demand growth is
winsorized at the 5th and 95th percentiles.} Beyond the benchmark sample,
we consider an extended sample spanning 1988Q4--2025Q4, which is the longest
period over which all twelve sectors are continuously observed.
\begin{table}[!t]
\caption{Benchmark Estimates of the Aggregate Equity Market Multiplier}
\label{tab:multiplier}
\centering
{\tmpsmall\sc \setlength{\tabcolsep}{6pt} \begin{threeparttable}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l cc cc@{}}
\toprule
& \multicolumn{2}{c}{1993Q1 -- 2018Q4} & \multicolumn{2}{c}{1988Q4 -- 2025Q4} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
& $n=12$ & $n=6$ & $n=12$ & $n=6$ \\
\midrule
OLS & $5.42^{***}$ & $10.05^{***}$ & $5.66^{***}$ & $11.42^{***}$ \\
& (0.43) & (0.78) & (0.52) & (1.11) \\
FIV & $4.42^{***}$ & $-7.13^{**}$ & $4.39^{***}$ & $-2.15$ \\
& (0.87) & (2.83) & (1.08) & (2.76) \\
GIV & $5.05^{***}$ & $8.70^{***}$ & $4.46^{***}$ & $9.42^{***}$ \\
& (0.30) & (0.82) & (0.30) & (0.93) \\
\midrule
$J$-test ($p$-value) & $< 0.001$ & 0.817 & $<0.001$ & 0.594 \\
Estimated $\bar{r}$ & 1 & 1 & 1 & 1 \\
$T$ & 104 & 104 & 149 & 149 \\
\bottomrule
\end{tabular*}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes.} The table reports estimates of the aggregate equity market multiplier, $\widehat{\Greekmath 0114 }=-1/\widehat{\Greekmath 011E }$, for the full twelve-sector panel ($n=12$) and the six-sector granular core ($n=6$). OLS is obtained from a regression of equally weighted demand, $n^{-1}\sum_{i\le n}\Delta q_{i,t}$, on the market return $\Delta p_t$. FIV denotes the factor-residual instrumental-variable estimator of \citet{gabaix2021search}, implemented using Algorithm~3 in Online Appendix~\ref{subsec:gk-replication} with two latent factors and four observed factors (GDP growth, size, value, and momentum). GIV denotes the demand-only granular-IV estimator based on the moment conditions in \eqref{Moment_phi}. The reported $J$-test $p$-value corresponds to the over-identification test, whose asymptotic null distribution is $\Greekmath 011F ^2_{n-\bar r-1}$, where $\bar r$ is selected using the BIC criterion in \eqref{BIC}--\eqref{r_hat}. Standard errors are reported in parentheses and are computed using the Newey--West estimator. For the OLS and GIV estimators, $\widehat{\Greekmath 0114 }$ is obtained from $\widehat{\Greekmath 011E }$ through the transformation $\widehat{\Greekmath 0114 }=-1/\widehat{\Greekmath 011E }$ and the corresponding standard errors are computed using the delta method. Significance levels correspond to two-sided tests of $H_0:\Greekmath 0114 =0$. Significance levels: $^{***}\,1\%$, $^{**}\,5\%$, $^{*}\,10\%$.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
For comparison, we also report estimates from the factor-residual IV (FIV) estimator of \citet{gabaix2021search}, together with the OLS estimator. The FIV estimator constructs instruments from estimated idiosyncratic demand shocks obtained after removing observed and latent demand factors.\footnote{Online Appendix~\ref{subsec:gk-replication} provides implementation details for the FIV estimator.} The OLS estimator is obtained from a regression of equally weighted demand, $n^{-1}\sum_{i\leq n}\Delta q_{i,t}$, on the market return $\Delta p_t$. Table~\ref{tab:multiplier} reports the resulting estimates and standard errors.
Across all specifications, the OLS estimate of the aggregate multiplier
exceeds its FIV and GIV counterparts. This pattern is consistent with the
endogeneity problem discussed earlier. Positive demand shocks increase both
holdings and prices, causing an uninstrumented regression to attribute part
of the demand shock to the price response. As a result, the demand
elasticity is biased toward zero and the implied multiplier is biased
upward. By exploiting granular demand variation that is orthogonal to common
demand factors, the GIV estimator corrects this source of bias.
When all twelve sectors are used to construct the GIVs, the estimated aggregate multiplier is $5.05$ in the benchmark sample 1993Q1--2018Q4 and $4.46$ in the extended sample 1988Q4--2025Q4, close to the corresponding FIV estimates. However, the over-identification test strongly rejects the associated moment restrictions in both samples, with $p$-values effectively equal to zero. One traditional interpretation of this result is that the instruments are invalid. Alternatively, following a perspective common in the treatment-effects literature, rejection of the over-identification test may reflect heterogeneity in the underlying causal parameters. From this perspective, the evidence suggests that the homogeneous-elasticity specification imposed on all twelve sectors is too restrictive and that demand elasticities may differ substantially across investor sectors.
To investigate this possibility, we restrict attention to the six largest
sectors, which together account for more than 97\% of total equity holdings
in the sample: households, mutual funds and ETFs, the foreign sector,
private pension funds, state and local pension funds, and life insurance
companies. Relative to the twelve-sector specification, both the OLS and GIV
estimates increase substantially. For example, the GIV estimate rises from $
5.05$ to $8.70$ in the benchmark sample and from $4.46$ to $9.42$ in the
extended sample. More importantly, the over-identification test no longer
rejects, yielding $p$-values of $0.817$ and $0.594$ in the two samples.
These findings suggest that the six largest sectors exhibit more homogeneous
demand behavior and therefore provide a more credible basis for estimating a
common aggregate demand elasticity and the corresponding market multiplier.
\begin{table}[!t]
\caption{Heterogeneous Demand Multipliers by Sector}
\label{tab:hetero}
\centering
{\tmpsmall\sc \setlength{\tabcolsep}{6pt} \begin{threeparttable}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l cc cc@{}}
\toprule
& \multicolumn{2}{c}{1993Q1--2018Q4} & \multicolumn{2}{c}{1988Q4--2025Q4} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}
Sector & $\hat{\Greekmath 0114 }$ & $J$-test ($p$-value) & $\hat{\Greekmath 0114 }$ & $J$-test ($p$-value) \\
\midrule
Granular core ($n=6$)
& $8.70^{***}$ & 0.817 & $9.42^{***}$ & 0.594 \\
& (0.82) & & (0.93) & \\
\addlinespace
Property and casualty insurers
& $3.57^{***}$ & 0.094 & $3.96^{***}$ & 0.927 \\
& (0.31) & & (0.38) & \\
\addlinespace
Federal government retirement funds
& $-18.15$ & 0.010 & $-17.32$ & $< 0.001$ \\
& (9.06) & & (12.99) & \\
\addlinespace
State and local governments
& $3.70^{***}$ & 0.419 & $3.65^{***}$ & 0.202 \\
& (0.45) & & (0.33) & \\
\addlinespace
Closed-end funds
& $6.63^{***}$ & 0.410 & $6.52^{***}$ & 0.356 \\
& (2.08) & & (1.71) & \\
\addlinespace
Banks
& $6.68^{***}$ & 0.347 & $10.68^{**}$ & 0.786 \\
& (1.70) & & (4.29) & \\
\addlinespace
Broker-dealers
& $-40.97$ & 0.064 & $-251.05$ & 0.011 \\
& (90.85) & & (3058.61) & \\
\bottomrule
\end{tabular*}
\begin{tablenotes}[flushleft]
\footnotesize
\item \textit{Notes.} The table reports sector-specific demand multipliers, $\hat{\Greekmath 0114 }_j=-1/\hat{\Greekmath 011E }_j$, estimated using the GIV moment conditions in \eqref{GIV_Form_n0_3}. The GIVs are constructed from the six largest equity-holding sectors (the granular core), which are assumed to share a common demand elasticity. For each sample period, the first row reproduces the corresponding six-sector estimate from Table~\ref{tab:multiplier}. Standard errors are reported in parentheses and are computed using the Newey--West estimator. The reported $p$-values correspond to the over-identification $J$-test. Rejection of the $J$-test indicates that the moment conditions constructed from the granular core are invalid for the corresponding sector. Standard errors for $\hat{\Greekmath 0114 }_j$ are computed using the delta method, and significance levels correspond to two-sided tests of $H_0:\Greekmath 0114 _j=0$. For sectors whose $J$-test rejects at the 5\% level, point estimates and standard errors are reported for completeness only, and statistical significance is not indicated. Significance levels: $^{***}\,1\%$, $^{**}\,5\%$, $^{*}\,10\%$.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
The comparison between the FIV and GIV estimators highlights the importance
of the fixed-$n$ approach. When all twelve sectors are included, the two
estimators deliver qualitatively similar conclusions. The FIV estimates of
the aggregate multiplier are $4.42$ in the benchmark sample and $4.39$ in
the extended sample, close to the corresponding GIV estimates of $5.05$ and $
4.46$, respectively, and broadly consistent with the estimates reported by
\citet{gabaix2021search}. Both instrumental-variable estimators also yield
smaller multipliers than OLS, as expected from the endogeneity bias
discussed above.
The contrast becomes much sharper when attention is restricted to the
six-sector granular core. In this case, the FIV estimator produces negative
multiplier estimates of $-7.13$ and $-2.15$, whereas the GIV estimator
yields stable and economically meaningful estimates of $8.70$ and $9.42$.
This divergence reflects the different identification strategies underlying
the two procedures. The FIV estimator constructs instruments from estimated
idiosyncratic demand shocks obtained after removing latent factors through
principal-components analysis and therefore relies on consistent estimation
of those factors. With only six sectors, the cross-sectional dimension is
too small for this approach to be reliable. By contrast, the fixed-$n$ GIV
estimator does not require consistent estimation of latent factors and
remains valid when the number of sectors is small. The resulting GIV
estimates are therefore more credible for the six-sector specification,
which is favored by the over-identification test and appears consistent with
the homogeneous-elasticity restriction underlying the aggregate multiplier.
Table \ref{tab:hetero} sheds light on the source of the rejection of the
twelve-sector specification. Using the GIVs constructed from the six-sector
core, we estimate sector-specific demand elasticities, $\Greekmath 011E _{j}$, and the
corresponding implied market multipliers, $\Greekmath 0114 _{j}\equiv-\Greekmath 011E _{j}^{-1}$,
for the remaining sectors. Several sectors exhibit substantially smaller
multipliers than the six-sector core. Property and casualty insurers and
state and local governments have multipliers between $3.5$ and $4.0$, less
than half the core estimate. Closed-end funds have intermediate multipliers
of roughly $6.5$, while banks exhibit larger and less precisely estimated
multipliers. For all of these sectors, the $J$-test provides little evidence
against the validity of the moment conditions constructed from the
six-sector core.
The two remaining sectors, federal government retirement funds and
broker-dealers, exhibit markedly different behavior. Their estimated multipliers are negative in both samples, implying non-positive estimates of the corresponding demand elasticities. Moreover, these estimates are highly
imprecise and economically difficult to interpret. More importantly, the $J$
-test rejects the validity of the moment conditions for federal government
retirement funds in both samples. For broker-dealers, the $J$-test is close
to rejection at the 5\% level in the shorter sample ($p$-value $=0.064$) and
is strongly rejected in the longer sample ($p$-value $=0.011$). Taken
together, these findings suggest that the moment conditions constructed from
the six-sector core do not provide valid identifying restrictions for these
two sectors. Consequently, the corresponding multiplier estimates should not
be given an economic interpretation.
Taken together, Tables ~\ref{tab:multiplier} and \ref{tab:hetero} provide
strong evidence that demand elasticities differ substantially across
investor sectors. Among the ten sectors for which the $J$-test does not
reject (treating broker-dealers as invalid given the decisive rejection in
the longer sample and the borderline $p$-value of $0.064$ in the shorter
sample), the estimated multipliers range from approximately $3.6$ to $10.7$,
with the upper end reflecting the imprecisely estimated bank sector. This
heterogeneity explains both the rejection of the twelve-sector specification
and the substantially smaller multiplier obtained when all sectors are
pooled together. By contrast, the six-sector core appears considerably more
homogeneous and yields a stable aggregate multiplier of roughly nine across
both sample periods. This estimate is an order of magnitude larger than the
frictionless benchmark multiplier of $0.05$ to $0.1$, providing evidence consistent with the inelastic-markets hypothesis.
\section{Conclusion\label{sec:conclusion}}
This paper develops an estimation and inference framework for structural
models identified by granular instrumental variables (GIVs). The key insight
is that, under suitable cross-sectional restrictions on the idiosyncratic
shocks, valid GIVs are characterized by the orthogonal complement of the
factor-loading space associated with latent aggregate shocks. This
characterization provides a transparent foundation for GIV-based
identification and allows demand and supply elasticities to be estimated
without conventional excluded instruments or direct observation of the
latent factors.
A central contribution of the paper is to show that the relevant orthogonal
complement can be identified and consistently estimated directly from the
covariance structure of the observables, without first estimating the latent
factors themselves. As a result, the proposed framework remains applicable
even when the number of entities is fixed and does not require the
cross-sectional dimension to diverge with the sample size. Building on this
result, we develop feasible procedures for estimation, inference, and
specification testing based on estimated GIVs. Monte Carlo evidence shows
that the feasible estimator performs similarly to an oracle procedure that
knows the true factor-loading space, while the empirical application yields
evidence consistent with highly inelastic aggregate equity demand.
The analysis also clarifies several identification issues that arise when
factor loadings are unknown. In particular, we show that certain
restrictions commonly imposed in existing implementations of the GIV
methodology should be interpreted as substantive assumptions on the latent
factors rather than innocuous normalizations. We further show that
identification can fail for existing moment-condition approaches when the
factor-loading space is unknown. The framework developed here avoids these
difficulties by focusing directly on the geometry of the factor-loading
space and exploiting the eigenspace structure of the covariance matrix of
the observables.
More broadly, the results demonstrate that cross-sectional heterogeneity can
be used not only to construct granular instruments but also to conduct valid
estimation, inference, and specification testing in models with latent
aggregate shocks. We hope that the framework developed in this paper will
facilitate the use of GIV methods in empirical work and provide a foundation
for future research in settings where latent aggregate forces and granular
heterogeneity interact.
\bigskip
{\tmpsmall\sc
\ifx\undefined\leavevmode\rule[.5ex]{3em}{.5pt}\
\fi
\ifx\undefined\textsc
\let\tmpsmall\tmpsmall\sc
\fi
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\end{thebibliography}
}
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